Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,13,16)(17,18,21,26)(19,28,30,23)(20,25,31,32)(22,24,29,27);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,21,30)(18,23,26,28)(20,22,31,29)(24,25,27,32);(9,11,14,10)(12,15,13,16)(18,24)(20,31)(22,29)(23,25)(26,27)(28,32);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20,21,31)(18,25,26,32)(19,29,30,22)(23,24,28,27);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(19,30)(20,31)(23,28)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7034362);