Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24778 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4)(9,10)(11,21)(12,25)(13,28)(14,32)(15,20)(16,23)(17,26)(18,30)(19,22)(24,38)(27,35)(29,34)(31,37)(33,40)(36,39);(5,6)(7,8)(9,11,15,24)(10,16,20,34)(12,14,27,22)(13,26,31,40)(17,32,33,19)(18,35,36,25)(21,37,38,28)(23,39,29,30);(2,4)(6,8);(9,12,15,27)(10,17,20,33)(11,22,24,14)(13,29,31,23)(16,19,34,32)(18,21,36,38)(25,37,35,28)(26,39,40,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,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, 408421);