Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,14,19,15,21,31,20)(11,22,24,37,26,39,25,33)(12,18,27,36,28,17,13,29)(16,32,34,40,35,30,38,23);(9,11)(10,16)(12,23)(13,30)(14,24)(15,26)(17,33)(18,37)(19,34)(20,38)(21,35)(22,29)(25,31)(27,32)(28,40)(36,39);(1,3,4,8,5,2,6,7)(9,12,14,27,15,28,31,13)(10,17,19,29,21,18,20,36)(11,23,24,32,26,40,25,30)(16,33,34,22,35,37,38,39);(2,3)(7,8)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391285);