Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1601 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)(3,7)(4,5)(6,8)(9,10,13,17,14,11,15,19)(12,18,21,24,20,16,22,23);(2,5)(6,7)(9,11,13,19,14,10,15,17)(12,18,22,23,20,16,21,24);(1,3,4,8)(2,6,5,7);(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^b=gi,e^d=ei,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257734, #72257750, #72257819, … |
| maximal normal subgroup of | #72257734, #72257750, #72257819, … |
| maximal quotient of | #72257954, #72264127, #72302336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300518);