Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(2,5)(6,7)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(3,8)(5,7)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gh,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,d^c=di,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232711, #72232718, #72232723, … |
| maximal normal subgroup of | #72232711, #72232718, #72232723, … |
| maximal quotient of | #72237747, #72237875, #72284858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34411);