Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,14,15,16)(17,18,20,22,21,24,19,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(18,22)(19,20)(23,24);(2,6)(5,7);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72309241, #72309359, #72309362 |
| maximal normal subgroup of | #72309241, #72309359, #72309362 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67684);