Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | #61316 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10,13,17,14,11,18,16)(12,15,21,23,20,19,24,22);(1,3,6,2)(4,7,5,8)(12,20)(15,19)(21,24)(22,23);(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,22)(23,24);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,12,13,21,14,20,18,24)(10,15,17,23,11,19,16,22)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72313678, #72315607, #72315643 |
| maximal normal subgroup of | #72313678, #72315607, #72315643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307928);