Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | |
2946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,19)(11,17,21,24)(12,22,20,16)(13,18,14,23);(5,6)(7,8)(9,11,15,21)(10,16,19,22)(12,13,20,14)(17,18,24,23);(2,4)(6,8);(2,4)(9,12,14,11,15,20,13,21)(10,17,18,22,19,24,23,16);(9,13,15,14)(10,18,19,23)(11,20,21,12)(16,24,22,17)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315330, #72315348, #72315350 |
| maximal normal subgroup of | #72315330, #72315348, #72315350 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409334);