Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,17,12,15)(11,16,20,24,21,23,19,22);(1,3)(2,5,6,7)(4,8)(10,15,17,18)(16,22,23,24);(9,11,13,20,14,21,12,19)(10,16,18,24,17,23,15,22);(10,17)(15,18)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,b^a=be,d^a=dh,e^a=eh,g^a=ghi,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72317647 |
| maximal normal subgroup of | #72317647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90048);