Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(25,26)(29,30);(1,3,5,7)(2,6,8,4)(9,11,12,15)(10,13,14,16)(17,18)(19,28)(20,26)(21,23)(22,25)(24,29)(27,32)(30,31);(1,4,5,6)(2,7,8,3)(10,14)(13,16)(17,19,21,27)(18,24,23,31)(20,29,22,30)(25,28,26,32);(2,8)(3,7)(10,14)(13,16)(17,20,21,22)(18,25,23,26)(19,29,27,30)(24,28,31,32);(18,23)(24,31)(25,26)(28,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411971);