Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,13,17,14,16,23,11)(12,19,21,24,20,15,22,18);(1,3,5,7,6,2,4,8)(9,11,13,16,14,17,23,10)(12,18,22,19,20,24,21,15);(9,12)(10,15)(11,18)(13,22)(14,20)(16,19)(17,24)(21,23);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103508);