Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,8,7)(9,10)(11,20)(12,15)(13,16)(14,28)(17,22)(18,24)(19,26)(21,33)(23,40)(25,36)(27,30)(29,35)(31,39)(32,37)(34,38);(1,2)(3,4)(5,7,8,6)(10,15)(11,21)(12,24)(14,26)(16,30)(17,33)(18,19)(20,32)(22,38)(25,37)(27,39)(28,31)(34,35)(36,40);(2,4)(6,7)(10,16)(15,30)(17,32)(18,31)(19,28)(20,33)(34,36)(35,40);(9,11,13,21)(10,17,16,32)(12,25,27,38)(14,29,26,23)(15,20,30,33)(18,35,31,40)(19,36,28,34)(22,39,37,24)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409378);