Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,24,21,25,29,22)(19,26,28,30,27,23,31,32);(10,14)(11,12)(18,22)(20,29)(23,30)(24,25)(26,32)(28,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,23,25,26)(20,28,29,31)(22,30,24,32);(9,12,13,11)(10,16,14,15);(1,3,4,7)(2,6,5,8)(19,27)(23,26)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375410);