Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #32904 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,20,23,21,24,19,22);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,22,24,23);(3,7)(5,8)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24);(3,7)(5,8)(9,11,12,15)(10,14,13,16);(2,6)(5,8)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542442);