Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510571 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31736 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,39,6,13,8,15)(3,10,37,58,18,44,20,46)(4,21,38,50,24,11,26,59)(5,27,30,51,31,84,40,12)(7,14,41,98,32,53,36,57)(16,42,70,107,62,100,63,101)(17,64,66,103,28,83,87,43)(19,45,92,116,67,104,33,54)(22,47,82,113,75,71,68,105)(23,76,79,48,77,111,91,72)(25,73,60,55,80,49,34,93)(29,85,90,56,35,95,97,52)(61,119,121,128,86,122,124,99)(65,114,96,106,69,117,88,102)(74,126,125,109,120,127,123,108)(78,118,89,110,81,115,94,112);(1,3,8,37,6,18,9,20)(2,10,39,46,13,44,15,58)(4,22,38,68,24,75,26,82)(5,28,40,87,31,17,30,66)(7,33,36,19,32,92,41,67)(11,47,50,113,21,71,59,105)(12,43,27,83,51,103,84,64)(14,54,98,104,53,116,57,45)(16,60,63,80,62,34,70,25)(23,74,79,123,77,120,91,125)(29,88,97,65,35,96,90,69)(42,55,107,73,100,93,101,49)(48,109,76,127,72,108,111,126)(52,114,85,102,56,117,95,106)(61,94,124,78,86,89,121,81)(99,118,119,112,128,115,122,110);(1,4,9,26,6,24,8,38)(2,11,15,59,13,21,39,50)(3,16,20,70,18,62,37,63)(5,23,40,79,31,77,30,91)(7,34,41,80,32,60,36,25)(10,42,58,101,44,100,46,107)(12,48,27,76,51,72,84,111)(14,55,57,49,53,93,98,73)(17,61,66,124,28,86,87,121)(19,68,33,22,67,82,92,75)(29,89,97,81,35,94,90,78)(43,99,83,119,103,128,64,122)(45,105,116,71,104,113,54,47)(52,115,85,110,56,118,95,112)(65,123,96,120,69,125,88,74)(102,127,117,108,106,126,114,109);(1,5)(2,12)(3,17)(4,23)(6,31)(7,35)(8,30)(9,40)(10,43)(11,48)(13,51)(14,56)(15,27)(16,61)(18,28)(19,69)(20,66)(21,72)(22,74)(24,77)(25,81)(26,79)(29,32)(33,88)(34,94)(36,97)(37,87)(38,91)(39,84)(41,90)(42,99)(44,103)(45,106)(46,64)(47,108)(49,112)(50,111)(52,53)(54,117)(55,118)(57,95)(58,83)(59,76)(60,89)(62,86)(63,121)(65,67)(68,125)(70,124)(71,109)(73,110)(75,120)(78,80)(82,123)(85,98)(92,96)(93,115)(100,128)(101,119)(102,104)(105,126)(107,122)(113,127)(114,116)Polycyclic presentation:
2^9|a^2=efh,b^2=fhi,c^2=ehi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417769);