Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,24)(19,27,28,23)(20,30,31,29)(21,25,32,26);(1,3,5,2)(4,7,8,6)(11,12)(14,16)(17,19,22,28)(18,23,24,27)(20,26,31,25)(21,30,32,29);(2,3)(6,7)(9,11,13,12)(10,14,15,16)(17,20,21,23,22,31,32,27)(18,19,26,30,24,28,25,29);(1,4,5,8)(2,6,3,7)(18,24)(19,28)(25,26)(29,30);(9,12,13,11)(10,14,15,16)(17,21,22,32)(18,25,24,26)(19,29,28,30)(20,23,31,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6705848);