Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,20,25,21,24,28,23)(19,22,27,31,26,30,32,29);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,22,24,30)(20,27,28,32)(23,29,25,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(9,11,12,15)(10,14,13,16)(19,26)(22,30)(27,32)(29,31);(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342848);