Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30400 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24296 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,16)(3,17,20,10)(4,24,30,83)(5,31,35,94)(6,36,42,62)(7,15,8,14)(11,49,54,27)(12,55,58,33)(13,40,63,41)(18,66,68,104)(19,71,32,95)(21,74,37,99)(22,47,23,48)(25,84,70,80)(26,51,90,114)(28,60,29,82)(34,98,43,61)(38,52,92,53)(39,64,102,57)(44,69,65,101)(45,97,56,72)(46,78,59,77)(50,75,105,86)(67,103,96,116)(73,117,79,107)(76,108,100,106)(81,123,126,88)(85,111,87,110)(89,115,93,113)(91,112,109,125)(118,119,121,127)(120,122,128,124);(1,3,8,23,9,20,7,22)(2,10,15,48,16,17,14,47)(4,25,29,18,30,70,28,68)(5,32,34,79,35,19,43,73)(6,37,41,77,42,21,40,78)(11,50,53,44,54,105,52,65)(12,56,57,108,58,45,64,106)(13,59,36,74,63,46,62,99)(24,80,60,66,83,84,82,104)(26,87,89,67,90,85,93,96)(27,75,38,101,49,86,92,69)(31,71,61,107,94,95,98,117)(33,72,39,76,55,97,102,100)(51,111,113,103,114,110,115,116)(81,124,125,118,126,122,112,121)(88,120,91,127,123,128,109,119);(1,4,9,30)(2,11,16,54)(3,18,20,68)(5,26,35,90)(6,38,42,92)(7,29,8,28)(10,44,17,65)(12,51,58,114)(13,60,63,82)(14,53,15,52)(19,67,32,96)(21,75,37,86)(22,70,23,25)(24,62,83,36)(27,40,49,41)(31,81,94,126)(33,91,55,109)(34,93,43,89)(39,88,102,123)(45,103,56,116)(46,80,59,84)(47,105,48,50)(57,115,64,113)(61,112,98,125)(66,99,104,74)(69,77,101,78)(71,121,95,118)(72,120,97,128)(73,87,79,85)(76,119,100,127)(106,111,108,110)(107,122,117,124);(1,5)(2,12)(3,19)(4,26)(6,39)(7,34)(8,43)(9,35)(10,45)(11,51)(13,61)(14,57)(15,64)(16,58)(17,56)(18,67)(20,32)(21,76)(22,73)(23,79)(24,81)(25,85)(27,91)(28,89)(29,93)(30,90)(31,36)(33,41)(37,100)(38,88)(40,55)(42,102)(44,103)(46,107)(47,106)(48,108)(49,109)(50,110)(52,113)(53,115)(54,114)(59,117)(60,112)(62,94)(63,98)(65,116)(66,118)(68,96)(69,120)(70,87)(71,99)(72,78)(74,95)(75,119)(77,97)(80,122)(82,125)(83,126)(84,124)(86,127)(92,123)(101,128)(104,121)(105,111)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416308);