Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,18)(12,23)(13,16)(14,17)(15,21)(19,22)(20,24);(1,3)(2,6)(4,7)(5,8)(9,10,13,23,14,17,12,16)(11,19,22,18,20,15,21,24);(2,7)(3,8)(11,20)(15,19)(18,24)(21,22);(1,4,5,6)(2,8,7,3)(9,10)(11,15)(12,16)(13,23)(14,17)(18,21)(19,20)(22,24);(9,11,13,22,14,20,12,21)(10,15,23,24,17,19,16,18)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7464577);