Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C23×C8 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,23,21,19,22,24);(1,2,3,4)(10,13)(14,15)(17,19,20,24,21,18,22,23);(9,11,12,16)(10,14,13,15);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321976, #72322076, #72322084 |
| maximal normal subgroup of | #72321976, #72322076, #72322084 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300493);