Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,16)(7,18)(8,24,27,9)(10,28,29,15)(12,26,14,17)(13,30)(19,23)(20,31)(21,36)(22,35)(25,33)(32,34);(1,2)(3,4)(5,7,9,20,12,23,29,34,11,22,24,30,14,33,10,21)(6,13,15,18,17,32,8,25,16,31,28,35,26,36,27,19);(2,4)(5,8,9,16,12,28,29,26,11,27,24,6,14,15,10,17)(7,19,20,13,23,18,34,32,22,25,30,31,33,35,21,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250558, #72250578, #72250583, … |
| maximal normal subgroup of | #72250558, #72250578, #72250583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56074);