Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| 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 |
| prime radical | #63050 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,9,23)(4,16,19,17)(5,10,22,27)(6,11,7,12)(15,28,24,31)(18,30,32,29)(20,25,21,26)(33,34,38,42)(35,43,46,56)(36,47,48,40)(37,50,52,41)(39,51,54,45)(44,58,59,57)(49,55,63,62)(53,60,61,64);(1,3,8,9)(2,4,13,19)(5,15,22,24)(6,16,7,17)(10,18,27,32)(11,14,12,23)(20,29,21,30)(25,31,26,28)(33,35,38,46)(34,39,42,54)(36,44,48,59)(37,51,52,45)(40,53,47,61)(41,56,50,43)(49,64,63,60)(55,58,62,57);(1,4)(2,9)(3,13)(5,18)(6,14)(7,23)(8,19)(10,24)(11,17)(12,16)(15,27)(20,31)(21,28)(22,32)(25,30)(26,29)(37,52)(41,50)(43,56)(45,51)(49,63)(55,62)(57,58)(60,64);(1,5)(2,10)(3,15)(4,18)(6,21)(7,20)(8,22)(9,24)(11,26)(12,25)(13,27)(14,28)(16,30)(17,29)(19,32)(23,31)(33,36)(34,40)(35,44)(37,49)(38,48)(39,53)(41,55)(42,47)(43,57)(45,60)(46,59)(50,62)(51,64)(52,63)(54,61)(56,58);(1,6,8,7)(2,11,13,12)(3,16,9,17)(4,14,19,23)(5,20,22,21)(10,25,27,26)(15,29,24,30)(18,31,32,28)(36,48)(40,47)(44,59)(49,63)(53,61)(55,62)(57,58)(60,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259212);