Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60907 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,14,19,15,18,13,24)(11,16,21,12,20,22,17,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,21,15,20,13,17)(10,16,19,12,18,22,24,23);(9,12,14,22,15,23,13,16)(10,17,19,11,18,21,24,20);(2,5)(6,8)(10,18)(11,20)(17,21)(19,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72315198, #72315641, #72315643 |
| maximal normal subgroup of | #72315198, #72315641, #72315643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6302510);