Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,7,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,20)(21,24)(22,23);(1,2,4,6,5,7,3,8)(9,11)(10,15,18,23)(12,20)(13,22)(14,21)(16,19,17,24);(10,16)(12,13)(15,19)(17,18)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239723, #72239729, #72239734, … |
| maximal normal subgroup of | #72239723, #72239729, #72239734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14217);