Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #45066 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,14,16,12)(19,26)(22,27)(28,32)(30,31);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,18,20,24)(19,27,26,22)(21,23,29,25)(28,31,32,30);(1,3,5,4)(2,7,8,6)(10,15)(11,16)(17,19,20,26)(18,22,24,27)(21,28,29,32)(23,30,25,31);(1,3,5,4)(2,7,8,6)(9,12,13,14)(10,16,15,11)(18,23)(21,29)(22,30)(24,25)(27,31)(28,32);(1,4,5,3)(2,7,8,6)(18,24)(19,26)(23,25)(28,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467361);