Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(9,10,12,11)(13,14)(15,19)(16,20)(17,18);(5,6)(7,8)(13,15,19,14,17,20,16,18);(1,2,3,4)(6,8)(14,18)(15,20);(9,11,12,10)(13,15,19,14,17,20,16,18)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fi,d^2=ehi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265052, #72265069, #72265098 |
| maximal normal subgroup of | #72265052, #72265069, #72265098 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284309);