Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #440901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15670 |
| 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 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,6,12,7,26)(3,14,17,32,18,9,28,41)(4,10,27,36,21,35,22,39)(5,23,29,52,24,11,25,38)(15,30,19,47,43,40,46,33)(16,37,44,56,45,31,54,61)(20,49,53,62,50,34,51,59)(42,55,48,58,63,60,64,57);(1,3,7,17,6,18,8,28)(2,9,26,41,12,14,13,32)(4,19,22,43,21,46,27,15)(5,16,25,44,24,45,29,54)(10,33,39,30,35,47,36,40)(11,37,52,56,23,31,38,61)(20,48,51,63,50,64,53,42)(34,58,62,60,49,57,59,55);(1,4,6,21)(2,10,12,35)(3,15,18,43)(5,20,24,50)(7,27,8,22)(9,30,14,40)(11,34,23,49)(13,39,26,36)(16,42,45,63)(17,46,28,19)(25,53,29,51)(31,55,37,60)(32,47,41,33)(38,62,52,59)(44,64,54,48)(56,58,61,57);(1,5)(2,11,12,23)(3,16)(4,20)(6,24)(7,25)(8,29)(9,31,14,37)(10,34,35,49)(13,38,26,52)(15,42)(17,44)(18,45)(19,48)(21,50)(22,51)(27,53)(28,54)(30,55,40,60)(32,56,41,61)(33,57,47,58)(36,59,39,62)(43,63)(46,64)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,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, 348098);