Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622850 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#447 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #3573 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,9)(3,14)(5,22)(6,7)(10,31)(11,29)(12,38)(13,30)(15,39)(16,43)(17,26)(19,46)(20,21)(23,27)(24,25)(28,33)(32,55)(34,53)(35,44)(36,54)(37,59)(41,49)(42,61)(45,48)(47,50)(52,57)(56,62)(58,64);(1,2,8,13)(3,10,17,35)(4,11,18,36)(5,23,25,38)(6,12,7,27)(9,24,30,22)(14,33,26,34)(15,32,41,56)(16,44,43,31)(19,47,45,59)(20,37,21,50)(28,51,53,40)(29,48,54,46)(39,57,49,58)(42,62,61,55)(52,63,64,60);(1,3)(2,10)(4,15)(5,16)(6,26)(7,14)(8,17)(9,28)(11,32)(12,34)(13,35)(18,41)(19,42)(20,49)(21,39)(22,40)(23,44)(24,51)(25,43)(27,33)(29,52)(30,53)(31,38)(36,56)(37,58)(45,61)(46,60)(47,62)(48,63)(50,57)(54,64)(55,59);(1,4)(2,11)(3,15)(5,19)(6,20)(7,21)(8,18)(9,29)(10,32)(12,37)(13,36)(14,39)(16,42)(17,41)(22,46)(23,47)(24,48)(25,45)(26,49)(27,50)(28,52)(30,54)(31,55)(33,57)(34,58)(35,56)(38,59)(40,60)(43,61)(44,62)(51,63)(53,64);(4,18)(11,36)(15,41)(19,45)(20,21)(29,54)(32,56)(37,50)(39,49)(42,61)(46,48)(47,59)(52,64)(55,62)(57,58)(60,63)Polycyclic presentation:
2^9|b^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,f^b=fh,h^b=hi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530048);