Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482075 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,15,17,24)(11,13)(12,20)(16,21,18,23)(19,22);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,21)(13,18)(14,17)(15,19)(20,24)(22,23);(2,6)(5,7);(9,11,14,13)(10,16,17,18)(12,19,22,20)(15,23,24,21)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279924, #72280040, #72280054, … |
| maximal normal subgroup of | #72279924, #72280040, #72280054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389274);