Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509855 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31064 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 128 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:
(1,2,9,39,35,60,36,61,8,15,6,13,7,14,43,65)(3,16,64,46,37,98,33,10,21,68,19,62,20,58,40,47)(4,22,24,83,25,52,92,99,27,53,41,101,26,11,48,51)(5,28,76,63,38,59,34,57,32,55,30,56,31,94,42,12)(17,69,71,49,72,108,100,104,74,115,116,44,73,66,90,81)(18,75,96,54,80,105,79,114,78,45,29,95,77,106,97,67)(23,84,86,50,87,110,102,111,89,124,119,112,88,93,91,82)(70,118,85,117,120,103,125,113,122,127,126,128,121,107,123,109);(1,3,8,21)(2,10,15,47)(4,17,27,74)(5,29,32,96)(6,33,9,40)(7,37,35,20)(11,49,52,44)(12,54,57,95)(13,58,39,98)(14,62,60,46)(16,65,68,61)(18,76,78,30)(19,43,64,36)(22,81,53,104)(23,85,89,126)(24,90,41,100)(25,73,26,72)(28,75,55,45)(31,97,38,79)(34,80,42,77)(48,71,92,116)(50,109,112,113)(51,69,99,115)(56,114,63,67)(59,106,94,105)(66,101,108,83)(70,119,122,86)(82,117,111,128)(84,118,124,127)(87,123,88,125)(91,120,102,121)(93,103,110,107);(1,4)(2,11)(3,17)(5,23)(6,24)(7,26)(8,27)(9,41)(10,44)(12,50)(13,51)(14,53)(15,52)(16,66)(18,70)(19,71)(20,73)(21,74)(22,60)(25,35)(28,93)(29,85)(30,86)(31,88)(32,89)(33,90)(34,91)(36,48)(37,72)(38,87)(39,99)(40,100)(42,102)(43,92)(45,103)(46,104)(47,49)(54,113)(55,110)(56,82)(57,112)(58,115)(59,84)(61,101)(62,81)(63,111)(64,116)(65,83)(67,117)(68,108)(69,98)(75,107)(76,119)(77,121)(78,122)(79,123)(80,120)(94,124)(95,109)(96,126)(97,125)(105,118)(106,127)(114,128);(1,5,8,32)(2,12,15,57)(3,18,21,78)(4,23,27,89)(6,34,9,42)(7,38,35,31)(10,45,47,75)(11,50,52,112)(13,59,39,94)(14,63,60,56)(16,67,68,114)(17,70,74,122)(19,79,64,97)(20,80,37,77)(22,82,53,111)(24,91,41,102)(25,88,26,87)(28,61,55,65)(29,40,96,33)(30,43,76,36)(44,103,49,107)(46,106,62,105)(48,86,92,119)(51,84,99,124)(54,98,95,58)(66,117,108,128)(69,109,115,113)(71,123,116,125)(72,121,73,120)(81,118,104,127)(83,93,101,110)(85,100,126,90)Polycyclic presentation:
2^9|a^2=efh,b^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defh,g^a=gi,c^b=cg,d^b=de,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417053);