Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(2,5)(6,7)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,28)(31,32);(9,12,13,16)(10,14,11,15);(10,11)(14,15)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(1,3,4,8)(2,6,5,7)(19,26)(22,30)(25,27)(29,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7452793);