Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4301 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)(3,6)(4,5)(7,8)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(10,15)(12,19)(14,20)(18,23);(1,3,4,7)(2,6,5,8);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72320791 |
| maximal normal subgroup of | #72320791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278151);