Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1171 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)(3,5)(4,7)(6,8)(9,10)(11,20,21,16)(12,17)(13,18,22,15)(14,19)(23,24);(2,5)(3,8)(6,7)(10,15)(11,21)(13,22)(16,17)(18,19)(20,24);(9,11)(10,16)(12,22)(13,23)(14,21)(15,24)(17,18)(19,20);(9,12,14,23)(10,17,19,24)(11,22,21,13)(15,16,18,20)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^c=fi,f^e=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389244);