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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,4,7)(9,10,12,14)(11,13,16,15)(17,18)(19,25)(20,24)(21,23)(22,27)(26,32)(28,30)(29,31);(3,4)(6,7)(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30);(1,3,5,4)(2,6,8,7)(10,13)(11,16)(14,15);(18,23)(22,25)(24,31)(30,32);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=h,e^2=fh,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, 6755445);