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 | |
2676 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)(11,12,16,18)(13,17,22,23)(19,20,24,21);(1,2,3,4)(9,11,14,16)(10,12,15,18)(13,21,22,20)(17,19,23,24);(6,8)(10,15)(11,18)(12,16)(17,23)(19,21)(20,24);(9,12,14,18)(10,16,15,11)(13,19,22,24)(17,20,23,21);(9,13,14,22)(10,17,15,23)(11,19,16,24)(12,20,18,21)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,d^b=di,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, 6481325);