Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,22)(19,23)(20,26)(24,28)(25,32)(27,31)(29,30);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(17,18,23,30,20,25,27,28,21,24,31,32,26,29,19,22);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(18,24)(22,28)(25,29)(30,32)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319253, #72320055, #72320093, … |
| maximal normal subgroup of | #72319253, #72320055, #72320093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532148);