Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×D42 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(2,6)(5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(10,13)(14,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(18,25)(19,29)(20,21)(23,31)(24,26)(27,32)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585634);