Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | #31070 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,15,16,14,19,13,18)(11,12,21,23,17,24,20,22);(1,3)(2,6)(4,7)(5,8)(9,11,14,17)(10,12,19,24)(13,20,15,21)(16,23,18,22);(10,16)(11,20)(13,15)(17,21)(18,19)(22,23);(2,7)(3,8)(9,12)(10,17)(11,19)(13,23)(14,24)(15,22)(16,21)(18,20);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^9|a^2=gi,b^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411139);