Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,19)(20,21)(23,24);(2,4)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,23)(22,24);(6,8)(17,19,21,18,22,23,20,24);(10,14)(13,16)Polycyclic presentation:
2^9|b^2=h,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321559, #72322093, #72322100 |
| maximal normal subgroup of | #72321559, #72322093, #72322100 |
| maximal quotient of | #72322963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317190);