Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,8,6,7,5,3)(9,10,12,17,13,15,21,23)(11,16,18,24,19,14,20,22);(1,3,5,7,6,8,4,2)(10,14,15,16)(17,24,23,22);(2,7)(3,8)(10,15)(14,16)(17,23)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=hi,b^a=bd,c^a=ch,d^a=dg,f^a=fi,g^a=gi,c^b=chi,e^b=egi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34345);