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 | |
2816 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)(9,10,15,18)(11,19,21,23)(12,13,17,14)(16,22,24,20);(5,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,22)(15,21)(17,23)(18,24);(2,4)(6,8);(2,4)(9,12,14,10,15,17,13,18)(11,16,20,19,21,24,22,23);(9,13,15,14)(10,17,18,12)(11,20,21,22)(16,23,24,19)Polycyclic presentation:
2^9|a^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, #72315347, #72315348 |
| maximal normal subgroup of | #72315330, #72315347, #72315348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409333);