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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 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)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,31,32,29);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,22)(20,27)(21,26)(23,31)(24,29)(25,30)(28,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(2,6)(7,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7474158);