Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4718 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)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23);(2,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=h,d^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72262941, #72262956, #72262999, … |
| maximal normal subgroup of | #72262941, #72262956, #72262999, … |
| maximal quotient of | #72303897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264661);