Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188968 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,18,20,25)(5,10)(6,11)(7,12)(8,15)(9,19,26,21)(14,22,29,32)(16,24)(17,23)(27,28,31,30)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,60,63)(36,46,47,56,48,40,49,54)(43,57,59,64,58,51,61,62);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,19,28,20,29,21,30)(9,22,25,27,26,32,18,31)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,52)(42,53)(46,57)(47,59)(48,58)(49,61)(50,60)(54,62)(55,63)(56,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,25)(12,26)(13,27)(15,28)(16,30)(17,29)(23,31)(24,32)(33,36,38,48)(34,40,41,46)(35,43,44,58)(37,49,50,47)(39,51,52,57)(42,56,55,54)(45,61,60,59)(53,64,63,62);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,18,26,25)(14,28,29,30)(22,31,32,27)(35,44)(39,52)(43,58)(45,60)(51,57)(53,63)(59,61)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,f^c=fh,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96205);