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 | |
1452 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)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,26)(20,30)(21,25)(22,24)(23,29)(27,32)(28,31);(2,5)(6,8)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(9,12,13,16)(10,14,11,15);(1,3,4,7)(2,6,5,8)(10,11)(14,15)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(2,5)(6,8)(17,20,21,22)(18,24,25,30)(19,28,29,27)(23,31,26,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7451480);