Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,19,14,16,11,17)(12,15,20,23,21,18,22,24);(10,15,16,18)(11,13)(17,24,19,23)(20,22);(2,5)(6,8)(10,16)(15,18)(17,19)(23,24);(9,11,14,13)(10,17,16,19)(12,20,21,22)(15,23,18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fgi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72278852, #72278872, #72278949, … |
| maximal normal subgroup of | #72278852, #72278872, #72278949, … |
| maximal quotient of | #72293127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387486);