Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(17,18)(19,28)(20,29)(21,27)(22,32)(23,26)(24,30)(25,31);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,25)(18,24,26,20)(21,30,22,29)(27,31,32,28);(10,13)(14,16)(17,20,23,24)(18,25,26,19)(21,31,22,28)(27,30,32,29);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,26)(19,25)(27,32)(29,30);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,27,26,32)(19,29,25,30)(20,31,24,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451099);