Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×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 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1138 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)(5,6,8,7)(9,10)(11,16)(12,22)(13,23)(14,18)(15,19)(17,20)(21,24);(2,4)(9,11,14,21,15,20,13,12)(10,16,18,24,19,17,23,22);(9,12,13,20,15,21,14,11)(10,17,18,22,19,16,23,24);(5,7,8,6)(11,20)(12,21)(16,17)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226894, #72226917 |
| maximal normal subgroup of | #72226894, #72226917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261868);