Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2679 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,24,20,22,27,29)(19,23,26,30,25,28,31,32);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,14,15);(2,6)(3,7)(11,16)(13,15)(18,22)(23,28)(24,29)(30,32);(2,6)(5,8)(10,14)(11,16);(17,19,21,26,20,25,27,31)(18,23,24,30,22,28,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfgi,d^b=df128 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340584 |
| maximal subgroup of | #72340548, #72340572, #72340584, … |
| maximal normal subgroup of | #72340548, #72340572, #72340584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315164);