Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
6260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,18,16,19,20,17);(1,2,3,4)(14,17)(15,20)(18,19);(6,8);(10,12)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72231457, #72239772, #72239776, … |
| maximal normal subgroup of | #72231457, #72239772, #72239776, … |
| maximal quotient of | #72231439, #72264616, #72281034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299819);