Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 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)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,28,32,26);(1,3)(2,5)(4,8)(6,7)(11,12)(14,15)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(2,6)(5,7);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(20,29)(24,32)(26,28)(27,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6754114);