Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
3830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(13,14,15,16)(18,22)(19,20)(23,24);(2,4)(9,10)(11,12)(17,18,20,23,21,24,19,22);(6,8)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf128 conjugacy classes — view the full character table.
| maximal subgroup of | #72314123, #72314156, #72314176, … |
| maximal normal subgroup of | #72314123, #72314156, #72314176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30707);