Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,24,23);(5,6)(7,8)(18,21)(19,24)(22,23);(2,4)(6,8);(9,11,12,16)(10,13,14,15)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264873, #72264880, #72264977, … |
| maximal normal subgroup of | #72264873, #72264880, #72264977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321825);