Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422137 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15302 |
| 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 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,17,9,18,34,30,35)(4,19,22,39,23,38,31,10)(5,24,27,56,28,11,26,41)(15,36,45,47,20,32,44,42)(16,40,25,33,46,54,58,61)(21,50,52,37,53,48,59,55)(43,62,51,63,49,60,57,64);(1,3,7,17,8,18,6,30)(2,9,12,34,13,35,29,14)(4,20,22,44,23,15,31,45)(5,25,27,46,28,58,26,16)(10,36,19,47,39,32,38,42)(11,40,41,33,24,54,56,61)(21,51,52,49,53,57,59,43)(37,63,48,60,55,64,50,62);(1,4,8,23)(2,10,13,39)(3,15,18,20)(5,21,28,53)(6,22,7,31)(9,32,35,36)(11,37,24,55)(12,38,29,19)(14,42,34,47)(16,43,46,49)(17,44,30,45)(25,57,58,51)(26,52,27,59)(33,60,61,62)(40,64,54,63)(41,50,56,48);(1,5,6,26,8,28,7,27)(2,11,12,41,13,24,29,56)(3,16,17,25,18,46,30,58)(4,21,22,52,23,53,31,59)(9,33,14,40,35,61,34,54)(10,37,38,50,39,55,19,48)(15,43,44,57,20,49,45,51)(32,60,42,64,36,62,47,63)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329334);