Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #95192 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
2442 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23,24,21)(20,22);(1,3)(2,5,6,7)(4,8)(10,14)(11,12)(15,16)(17,19)(18,21)(20,24)(22,23);(9,11,13,12)(10,14,16,15)(18,22)(19,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gi,d^2=gi,b^a=bd,c^a=cg,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240349, #72240353, #72260925, … |
| maximal normal subgroup of | #72240349, #72240353, #72260925, … |
| maximal quotient of | #72240351, #72261314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2405);