Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1858 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)(3,4)(9,10,12,16,14,17,13,22)(11,18,20,15,21,23,19,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,24,22,23);(2,4)(5,6,7,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,g^a=gi,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226702 |
| maximal normal subgroup of | #72226702 |
| maximal quotient of | #72226698 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30624);