Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(10,14)(11,15)(16,21)(17,22)(18,20)(23,24);(1,2)(3,4)(6,7)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24);(2,4)(5,6,8,7)(9,11,14,20,13,18,10,15)(12,17,21,24,19,23,16,22);(2,4)(5,7,8,6)(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24);(12,19)(16,21)(17,23)(22,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307324, #72307447, #72307652 |
| maximal normal subgroup of | #72307324, #72307447, #72307652 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351452);