Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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 | C23×C22⋊C4 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,16,12,21)(11,18,19,15,20,24,22,23);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11)(10,15)(12,22)(13,19)(14,20)(16,23)(17,24)(18,21);(2,6)(5,7);(10,16)(15,23)(17,21)(18,24)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235088, #72235098, #72235099, … |
| maximal normal subgroup of | #72235088, #72235098, #72235099, … |
| maximal quotient of | #72237203, #72237222, #72238889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281423);