Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453828 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16135 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
625 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)(3,8,15,31)(4,17,21,43)(5,22)(6,12)(7,11)(9,20,33,47)(10,26)(13,37)(14,29,23,50)(16,30,27,36)(18,41)(19,46,49,42)(24,35)(25,52)(28,45)(32,58,51,48)(34,54)(38,60)(39,56)(40,57,53,59)(44,62)(55,63)(61,64);(1,3)(2,8,11,31)(4,18)(5,23,26,14)(6,27)(7,15)(9,28,33,56)(10,29)(12,36,35,30)(13,21)(16,24)(17,41,43,37)(19,38,49,44)(20,39)(22,50)(25,40,54,53)(32,55)(34,59)(42,62)(45,47)(46,60)(48,63,58,61)(51,64)(52,57);(1,4)(2,9,11,33)(3,13,15,18)(5,19)(6,20)(7,21)(8,28)(10,32,22,51)(12,17,35,43)(14,38,23,44)(16,39,27,45)(24,47)(25,48)(26,49)(29,55)(30,37)(31,56)(34,42,52,46)(36,41)(40,61,53,63)(50,64)(54,58)(57,60)(59,62);(1,5,24,54,7,26,6,25)(2,10,12,34,11,22,35,52)(3,14,16,40,15,23,27,53)(4,19,47,58,21,49,20,48)(8,29,36,59,31,50,30,57)(9,32,17,42,33,51,43,46)(13,38,39,61,18,44,45,63)(28,55,41,62,56,64,37,60)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=e,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361025);