Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | #35165 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1515 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)(7,13,16,20)(8,12,14,18)(9,11)(10,15)(17,19);(5,7,14,15)(6,10,18,20)(8,17,9,16)(11,19,12,13);(1,2,3,4);(6,11)(10,19)(12,18)(13,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,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 | #72263571, D4×#2127, #72278372, … |
| maximal normal subgroup of | #72263571, D4×#2127, #72278372, … |
| maximal quotient of | #72318034, #72318038 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402369);