Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,20,24,21,25,28,31,22,19,23,26,29,27,30,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9);(17,19,20,26,21,27,28,32,22,18,23,24,29,25,30,31);(2,7)(6,15)(8,12)(9,16)(18,19)(24,26)(25,27)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,d^a=dgi,e^a=eg,e^b=eg,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72321224 |
| maximal normal subgroup of | #72321224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390855);