Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13,15,9)(8,16)(10,12)(11,18)(14,17,20,19);(5,7,13,12)(6,10,15,9)(8,14,19,16)(11,18,20,17);(1,2,3,4);(5,8,10,18)(6,11,12,16)(7,14,15,20)(9,17,13,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280656, #72312822, #72312831, … |
| maximal normal subgroup of | #72280656, #72312822, #72312831, … |
| maximal quotient of | #72280711, C2×#495174 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402370);