Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45353 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16);(2,6)(7,8)(11,16)(13,15)(17,18,22,26)(19,23,27,32)(20,25,21,24)(28,31,29,30);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,29)(21,28)(22,27)(24,31)(25,30)(26,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32);(2,6)(7,8)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,31,32,30)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=f,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,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257516, #72257535, #72264215 |
| maximal normal subgroup of | #72257516, #72257535, #72264215 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7440240);