Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351787 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,20)(14,22)(15,24)(17,23)(19,25)(21,26)(27,29)(28,31)(30,32)(33,34)(35,45)(36,41)(37,44)(38,42)(39,43)(40,47)(46,60)(48,58)(49,57)(50,55)(51,56)(52,54)(53,61)(59,62)(63,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,13)(11,23)(12,24)(18,31)(19,28)(20,29)(21,30)(25,32)(26,27)(33,35)(34,40)(36,46)(37,47)(38,52)(39,48)(41,53)(42,45)(43,54)(44,58)(49,61)(50,63)(51,62)(55,60)(56,64)(57,59);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,24,32)(10,26,11,18)(13,27,23,31)(15,29,16,28)(33,36,39,51)(34,41,43,56)(35,46,48,62)(37,50,38,49)(40,53,54,64)(42,57,44,55)(45,59,58,60)(47,63,52,61);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,25,26)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,42,43,44)(35,47,48,52)(36,49,51,50)(40,45,54,58)(41,55,56,57)(46,61,62,63)(53,60,64,59);(1,5,7,6)(2,11,12,10)(3,15,17,16)(4,20,21,19)(8,23,24,13)(9,18,25,26)(14,29,30,28)(22,31,32,27)(34,43)(40,54)(41,56)(42,44)(45,58)(53,64)(55,57)(59,60)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259012);