Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #2699 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26,22,28,20,25)(19,29,30,32,27,24,31,21);(1,2)(3,7)(4,8)(5,6)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,26)(22,27)(23,30)(25,32)(28,29);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,22,23)(18,25,28,26)(19,30,27,31)(21,29,32,24);(1,4,5,3)(2,7,6,8)(17,20,22,23)(18,26,28,25)(19,30,27,31)(21,24,32,29);(17,21,23,29,22,32,20,24)(18,27,26,31,28,19,25,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ci,d^b=df,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292845);