Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394177 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,12,14)(11,15,16,13)(18,19)(20,21)(23,24);(1,2)(3,4)(9,11,12,16)(10,13,14,15)(17,18,20,19,22,24,21,23);(2,4)(17,19,21,18,22,23,20,24);(6,8)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239681, #72239691, #72239782 |
| maximal normal subgroup of | #72239681, #72239691, #72239782 |
| maximal quotient of | #72250540 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305871);