Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,12,17,14,18,13,16)(11,19,21,15,20,24,22,23);(5,6,8,7)(9,11,13,22,14,20,12,21)(10,15,17,24,18,23,16,19);(2,4);(5,7,8,6)(11,20)(15,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239131, #72239140, #72239146, … |
| maximal normal subgroup of | #72239131, #72239140, #72239146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262122);