Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480268 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,15,16,10)(9,12)(13,18)(14,20,19,17);(6,10)(11,17)(12,15)(18,20);(2,4)(7,13)(11,18)(14,19)(17,20);(5,7,9,13)(6,11,12,18)(8,14,16,19)(10,17,15,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219302, #72225779, #72232794, … |
| maximal normal subgroup of | #72219302, #72225779, #72232794, … |
| maximal quotient of | #72232782, #72233220, C2×#493425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400621);