Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #406106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14924 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 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)(3,14)(4,9)(5,24,26,36)(6,12)(7,13)(8,17)(10,25,37,55)(11,27)(15,39)(16,29,45,57)(18,31)(19,47)(20,49,51,54)(21,38)(22,34)(23,35)(28,42)(30,46)(32,43)(33,52,59,50)(40,44,53,63)(41,58,48,56)(60,62,64,61);(1,3,7,18)(2,8)(4,19)(5,16,26,45)(6,17,27,46)(9,32,35,28)(10,29)(11,14)(12,31)(13,30)(15,23)(20,48)(21,43)(22,42)(24,53)(25,44,55,63)(33,58,59,56)(34,47,38,39)(36,40)(37,57)(41,51)(49,64,54,60)(50,62)(52,61);(1,4,7,23)(2,9,13,35)(3,15,18,19)(5,20,26,51)(6,21,27,22)(8,28,30,32)(10,33,37,59)(11,34,12,38)(14,39,31,47)(16,41,45,48)(17,42,46,43)(24,54,36,49)(25,50,55,52)(29,56,57,58)(40,60,53,64)(44,61,63,62);(1,5,6,25,7,26,27,55)(2,10,11,36,13,37,12,24)(3,16,17,44,18,45,46,63)(4,20,21,50,23,51,22,52)(8,29,14,40,30,57,31,53)(9,33,34,49,35,59,38,54)(15,41,42,61,19,48,43,62)(28,56,39,60,32,58,47,64)Polycyclic presentation:
2^9|a^2=f,b^2=eh,c^2=i,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313303);