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 | |
1673 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)(24,31)(26,32);(1,3)(2,5)(4,8)(6,7)(17,18)(19,26)(20,22)(21,23)(24,30)(25,29)(27,32)(28,31);(9,11,12,16)(10,13,14,15)(18,23)(21,29)(22,25)(24,32)(26,31)(28,30);(2,6)(5,7)(19,27)(24,31)(26,32)(28,30);(3,8)(5,7)(17,19,20,27)(18,24,22,31)(21,30,29,28)(23,32,25,26)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=i,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 | #72285234, #72285503 |
| maximal normal subgroup of | #72285234, #72285503 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595265);