Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,24,21,25,28,32)(19,22,27,29,26,23,30,31);(1,3,4,8)(2,5,6,7)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,30)(31,32);(3,8)(5,7)(10,13)(14,16);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32);(19,26)(22,23)(27,30)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282349);