Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,24,21)(22,23);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,21)(13,20)(15,23)(16,18)(17,24)(19,22);(9,11)(10,15)(12,21)(13,20)(14,17)(16,24)(18,23)(19,22);(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120970);