Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359617 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13,16,8)(4,9)(5,22,24,10)(6,25)(7,11)(12,28)(14,29,42,39)(15,30)(17,31,26,37)(18,46,45,32)(19,48,50,33)(20,38)(21,34)(23,36,54,51)(27,35)(40,43)(41,61)(44,59)(47,57)(49,58,53,60)(52,56)(55,62)(63,64);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,32)(10,30)(11,13)(12,37)(14,20)(17,28)(19,47)(21,45)(22,40)(23,52)(24,43)(25,31)(27,42)(29,35)(33,57)(34,46)(36,59)(38,39)(41,49)(44,54)(48,64)(50,63)(51,56)(53,62)(55,58)(60,61);(1,4,7,21)(2,9,11,34)(3,14,16,42)(5,19,24,50)(6,27,28,20)(8,29,13,39)(10,33,22,48)(12,38,25,35)(15,41,43,62)(17,45,26,18)(23,53,54,49)(30,55,40,61)(31,46,37,32)(36,60,51,58)(44,63,52,47)(56,64,59,57);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,30)(9,33)(11,22)(12,36)(13,40)(14,41)(16,43)(17,44)(18,47)(20,49)(21,50)(25,51)(26,52)(27,53)(28,54)(29,55)(31,56)(32,57)(34,48)(35,58)(37,59)(38,60)(39,61)(42,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266814);