Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(11,15)(12,18)(16,20)(17,24)(19,23)(21,22);(9,10,15,22,12,17,19,20,13,16,23,24,18,21,11,14);(2,4)(10,16)(14,20)(17,21)(22,24);(5,7,8,6)(10,16)(14,20)(17,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72344012, #72344017 |
| maximal normal subgroup of | #72344012, #72344017 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403069);