Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,38)(3,18,23,52)(4,6,14,30)(5,13,39,49)(7,26,33,48)(8,16,9,17)(11,40,44,22)(12,28,34,27)(15,37,64,29)(19,54,43,20)(21,51,62,55)(24,50,25,53)(31,47,32,36)(35,45,61,46)(41,63,58,42)(56,60,57,59);(1,3)(2,11)(4,19)(5,20)(6,21)(7,35)(8,24)(9,25)(10,23)(12,41)(13,42)(14,43)(15,50)(16,45)(17,46)(18,26)(22,29)(27,56)(28,57)(30,62)(31,59)(32,60)(33,61)(34,58)(36,51)(37,40)(38,44)(39,54)(47,55)(48,52)(49,63)(53,64);(1,4,8,27)(2,12,16,6)(3,19,24,56)(5,15,31,26)(7,36,29,49)(9,28,10,14)(11,41,45,21)(13,33,47,37)(17,30,38,34)(18,20,50,59)(22,63,35,51)(23,43,25,57)(32,48,39,64)(40,42,61,55)(44,58,46,62)(52,54,53,60);(1,5)(2,13)(3,20)(4,26)(6,33)(7,30)(8,31)(9,32)(10,39)(11,42)(12,37)(14,48)(15,27)(16,47)(17,36)(18,19)(21,61)(22,58)(23,54)(24,59)(25,60)(28,64)(29,34)(35,62)(38,49)(40,41)(43,52)(44,63)(45,55)(46,51)(50,56)(53,57);(1,6,10,30)(2,14,38,4)(3,21,23,62)(5,29,39,37)(7,32,33,31)(8,12,9,34)(11,43,44,19)(13,15,49,64)(16,28,17,27)(18,51,52,55)(20,22,54,40)(24,41,25,58)(26,36,48,47)(35,60,61,59)(42,50,63,53)(45,57,46,56)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259654);