Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3282 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)(6,7)(9,10,15,11)(12,16,22,18)(13,19,14,17)(20,21,24,23);(5,6,8,7)(9,11,15,10)(12,20,22,24)(13,19,14,17)(16,21,18,23);(2,4)(5,7,8,6)(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,20)(19,24);(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,20)(19,24);(9,13,15,14)(10,17,11,19)(12,21,22,23)(16,20,18,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345853);