Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,12,15)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,3,4,8)(2,5,6,7)(11,12)(14,15);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(2,6)(5,7);(9,12,13,11)(10,15,16,14)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6755782);