Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,6,8,5,3)(9,10,13,18,14,19,11,16)(12,17,20,24,21,15,22,23);(1,3,5,8,6,7,4,2)(10,15,19,17)(16,23,18,24);(9,11,14,13)(10,16,19,18)(12,20,21,22)(15,23,17,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=h,f^2=hi,b^a=bd,d^a=dg,f^a=fi,g^a=gi,e^b=egi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34274);