Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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×#548 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4530 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)(3,7,8,5)(4,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24);(10,18)(15,19)(16,17)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72339503, #72339504, #72339512, … |
| maximal normal subgroup of | #72339503, #72339504, #72339512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293487);