Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342574 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4497 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)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(11,18)(14,22)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=h,d^2=f,b^a=bf,c^a=cg,g^a=gi,e^b=ei,h^c=hi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72252422, #72252426, #72252427 |
| maximal normal subgroup of | #72252422, #72252426, #72252427 |
| maximal quotient of | #72267128 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278140);