Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,16,13,18)(11,15,20,23,19,22,21,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,19,12,20)(10,15,18,24,16,22,17,23);(11,19)(15,22)(20,21)(23,24);(10,16)(15,22)(17,18)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=be,d^a=di,e^a=eh,c^b=ci,f^b=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235961, #72235984, #72235990, … |
| maximal normal subgroup of | #72235961, #72235984, #72235990, … |
| maximal quotient of | #72264327, #72311032, #72314235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88676);