Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 |
| prime radical | #62902 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,15)(3,18,38,63)(4,5,26,12)(6,14,33,47)(7,35,34,48)(9,16,10,17)(11,39,49,21)(13,31,27,30)(19,53,62,57)(20,56,40,54)(22,42,23,52)(24,28,25,32)(29,37,45,46)(36,43,60,44)(41,64,55,50)(51,59,61,58);(1,3)(2,11)(4,19)(5,20)(6,21)(7,36)(8,38)(9,22)(10,23)(12,40)(13,41)(14,42)(15,49)(16,43)(17,44)(18,35)(24,51)(25,61)(26,62)(27,55)(28,57)(29,50)(30,58)(31,59)(32,53)(33,39)(34,60)(37,56)(45,64)(46,54)(47,52)(48,63);(1,4)(2,12)(3,19)(5,15)(6,24)(7,37)(8,26)(9,27)(10,13)(11,40)(14,32)(16,31)(17,30)(18,50)(20,49)(21,51)(22,55)(23,41)(25,33)(28,47)(29,35)(34,46)(36,56)(38,62)(39,61)(42,53)(43,59)(44,58)(45,48)(52,57)(54,60)(63,64);(1,5,9,30)(2,13,16,26)(3,20,22,58)(4,15,27,17)(6,32,7,29)(8,12,10,31)(11,41,43,62)(14,37,35,24)(18,51,42,56)(19,49,55,44)(21,53,36,50)(23,59,38,40)(25,47,46,48)(28,34,45,33)(39,57,60,64)(52,54,63,61);(1,6,9,7)(2,14,16,35)(3,21,22,36)(4,24,27,37)(5,28,30,45)(8,33,10,34)(11,42,43,18)(12,32,31,29)(13,46,26,25)(15,47,17,48)(19,51,55,56)(20,57,58,64)(23,60,38,39)(40,53,59,50)(41,54,62,61)(44,63,49,52)Polycyclic presentation:
2^9|a^2=hi,d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260440);