Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,15)(11,19,21,16)(12,17,22,20)(13,18,23,24);(1,2,3,4)(9,11)(10,16)(12,13)(14,21)(15,19)(17,24)(18,20)(22,23);(6,8)(9,10)(12,22)(13,18)(14,15)(16,19)(23,24);(9,12,14,22)(10,17,15,20)(11,13,21,23)(16,24,19,18);(9,13,14,23)(10,18,15,24)(11,20,21,17)(12,16,22,19)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481022);