Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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 | |
1100 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)(35,39,44,53)(36,46,48,63)(37,50,51,41)(40,55,57,49)(43,58,47,62)(45,61,60,54)(52,64,56,59);(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)(33,35,38,44)(34,39,42,53)(36,47,48,43)(37,45,51,60)(40,56,57,52)(41,54,50,61)(46,62,63,58)(49,64,55,59);(1,4,7,3)(2,8,10,11)(5,12,6,13)(9,16,14,15)(18,23)(19,26)(24,28)(27,32)(33,36,37,49,38,48,51,55)(34,40,41,46,42,57,50,63)(35,43,45,59,44,47,60,64)(39,52,54,58,53,56,61,62);(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)(35,44)(36,48)(39,53)(40,57)(45,60)(46,63)(49,55)(54,61);(1,6,7,5)(2,9,10,14)(3,12,4,13)(8,16,11,15)(35,44)(39,53)(43,47)(45,60)(52,56)(54,61)(58,62)(59,64)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893185);