Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239031, #72239034, #72239035, … |
| maximal normal subgroup of | #72239031, #72239034, #72239035, … |
| maximal quotient of | #72224705, #72239309, #72239578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279667);