Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1738 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(18,22)(23,25)(24,29)(30,31);(1,3)(2,5)(4,8)(6,7)(17,18)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,32);(9,11,12,16)(10,13,14,15)(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(2,6)(5,7)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(3,8)(5,7)(18,25)(19,26)(21,28)(22,23)(24,30)(29,31)Polycyclic presentation:
2^9|a^2=g,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285198 |
| maximal normal subgroup of | #72285198 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595264);