Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632008 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,20,24,21,25,28,23)(19,22,26,30,27,31,32,29);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,13,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(3,7)(5,8)(9,12,14,13)(10,15,11,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(3,7)(5,8)(9,13,14,12)(10,15,11,16);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542794);