Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1344 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)(11,14)(12,15)(13,16)(19,26)(20,21)(23,31)(24,29)(27,32)(28,30);(1,3,5,8)(2,6,4,7)(9,11,14,10)(12,15,13,16)(17,18)(19,27)(20,29)(21,24)(22,25)(23,28)(26,32)(30,31);(9,12)(10,15)(11,16)(13,14)(17,19,20,30,22,28,21,26)(18,23,29,32,25,27,24,31);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,16,11,15)(17,18)(19,23)(20,29)(21,24)(22,25)(26,31)(27,28)(30,32);(10,11)(15,16)(17,20,22,21)(18,24,25,29)(19,26,28,30)(23,32,27,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=gi,d^2=f,e^2=i,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6950700);