Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8610 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(2,5)(3,7)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239050, #72239111 |
| maximal normal subgroup of | #72239050, #72239111 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13310);