Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1890 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18,21,20,22,25,23,26)(19,27,29,32,30,31,28,24);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,25,32)(20,31,26,27)(21,28,23,29);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,18,23,26,22,25,21,20)(19,27,28,24,30,31,29,32);(1,4,5,3)(2,8,6,7)(17,20,21,25,22,26,23,18)(19,24,29,27,30,32,28,31);(17,21,22,23)(18,20,25,26)(19,28,30,29)(24,31,32,27)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365745);