Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,13,14,12,16,11,15)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29);(2,5)(6,8)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31);(9,10,12,16)(11,14,13,15)(19,26)(22,25)(27,32)(29,31);(1,3,4,7)(2,6,5,8)(9,11,12,13)(10,14,16,15)(20,28)(23,30)(27,32)(29,31);(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451505);