Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508032 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29594 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
952 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,16)(4,23)(5,12)(6,33)(7,14)(8,39)(9,15)(10,19)(11,26)(13,36)(17,59)(18,56)(20,69)(21,58)(22,43)(24,78)(25,76)(27,52)(28,87)(29,77)(30,88)(31,49)(32,51)(34,54)(35,53)(37,75)(38,50)(40,94)(41,65)(42,71)(44,70)(45,81)(46,84)(47,85)(48,90)(55,62)(57,72)(60,110)(61,96)(63,74)(64,113)(66,116)(67,100)(68,92)(73,99)(79,124)(80,101)(82,122)(83,123)(86,106)(89,103)(91,107)(93,108)(95,114)(97,111)(98,120)(102,119)(104,126)(105,125)(109,118)(112,121)(115,127)(117,128);(1,3,8,22,9,19,7,21)(2,10,14,43,15,16,39,58)(4,24,28,17,29,63,27,61)(5,18,31,67,32,65,38,73)(6,20,36,72,37,68,35,71)(11,44,46,55,47,101,54,40)(12,41,50,100,51,56,49,99)(13,42,33,92,53,57,75,69)(23,74,52,59,77,78,87,96)(25,79,82,60,83,112,86,64)(26,80,34,62,85,70,84,94)(30,66,90,118,91,115,93,120)(45,102,104,97,105,128,103,95)(48,98,88,127,108,109,107,116)(76,121,106,110,123,124,122,113)(81,117,89,111,125,119,126,114);(1,4,9,29)(2,11,15,47)(3,17,19,61)(5,25,32,83)(6,34,37,84)(7,28,8,27)(10,40,16,55)(12,45,51,105)(13,52,53,87)(14,54,39,46)(18,64,65,60)(20,70,68,80)(21,63,22,24)(23,75,77,33)(26,35,85,36)(30,89,91,126)(31,86,38,82)(41,97,56,95)(42,74,57,78)(43,101,58,44)(48,106,108,122)(49,104,50,103)(59,92,96,69)(62,71,94,72)(66,117,115,119)(67,112,73,79)(76,107,123,88)(81,93,125,90)(98,124,109,121)(99,128,100,102)(110,116,113,127)(111,118,114,120);(1,5,9,32)(2,12,15,51)(3,18,19,65)(4,25,29,83)(6,30,37,91)(7,38,8,31)(10,41,16,56)(11,45,47,105)(13,48,53,108)(14,50,39,49)(17,60,61,64)(20,66,68,115)(21,73,22,67)(23,76,77,123)(24,79,63,112)(26,81,85,125)(27,86,28,82)(33,88,75,107)(34,89,84,126)(35,93,36,90)(40,95,55,97)(42,98,57,109)(43,100,58,99)(44,102,101,128)(46,104,54,103)(52,106,87,122)(59,110,96,113)(62,111,94,114)(69,116,92,127)(70,119,80,117)(71,120,72,118)(74,121,78,124)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fh,g^c=gi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415231);