Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1464 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)(18,23)(19,26)(21,28)(24,31)(27,29)(30,32);(1,3,5,8)(2,6,4,7)(10,11)(12,13)(17,18,22,23)(19,27,30,24)(20,28,25,21)(26,31,32,29);(1,2,5,4)(3,6,8,7)(9,11,14,10)(12,15,13,16)(17,19,28,32,22,30,21,26)(18,24,25,31,23,27,20,29);(1,4,5,2)(3,6,8,7)(9,12,14,13)(10,15,11,16)(17,20,22,25)(18,21,23,28)(19,29,30,31)(24,26,27,32);(9,13,14,12)(10,15,11,16)(17,21,22,28)(18,25,23,20)(19,26,30,32)(24,31,27,29)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7094658);