Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,7)(14,19)(15,20)(16,17);(9,10,11,12)(13,14,17,15,18,20,16,19);(2,4)(13,15,16,14,18,19,17,20);(5,7,8,6)(14,20)(15,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239687, #72239718, #72239733, … |
| maximal normal subgroup of | #72239687, #72239718, #72239733, … |
| maximal quotient of | #72250722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261573);