Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,20)(21,23)(22,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,24);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,20,18,21,24,22,23);(2,6)(5,7)(11,16)(13,15)(17,20,21,22)(18,24,23,19);(10,14)(13,15)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816196);