Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,10)(3,11,4,8)(5,14,6,9)(12,15,13,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,29)(27,30)(31,32)(33,34,38,42)(35,43,46,39)(36,47,50,54)(37,51,52,41)(40,48,56,44)(45,53,59,57)(49,62,63,60)(55,58,64,61);(1,3,7,4)(2,8,10,11)(5,12,6,13)(9,15,14,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,38,46)(34,39,42,43)(36,48,50,44)(37,45,52,59)(40,54,56,47)(41,53,51,57)(49,61,63,58)(55,62,64,60);(1,4,7,3)(2,8,10,11)(5,12,6,13)(9,16,14,15)(18,23)(19,26)(24,28)(27,32)(33,36,35,44,38,50,46,48)(34,40,39,47,42,56,43,54)(37,49,45,58,52,63,59,61)(41,55,53,60,51,64,57,62);(1,5,7,6)(2,9,10,14)(3,12,4,13)(8,15,11,16)(17,20,21,29)(18,24,23,28)(19,27,26,32)(22,30,25,31)(33,37)(34,41)(35,45)(36,49)(38,52)(39,53)(40,55)(42,51)(43,57)(44,58)(46,59)(47,60)(48,61)(50,63)(54,62)(56,64);(1,6,7,5)(2,9,10,14)(3,12,4,13)(8,16,11,15)(37,52)(41,51)(45,59)(49,63)(53,57)(55,64)(58,61)(60,62)Polycyclic presentation:
2^9|a^2=fhi,b^2=fhi,c^2=fgi,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893293);