Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,28)(29,31)(30,32);(1,3,4,8)(2,5,6,7)(18,23)(19,27)(20,28)(22,30)(24,29)(25,31);(2,6)(5,7)(10,13)(14,16)(17,19,21,25)(18,24,23,20)(22,31,30,27)(26,29,32,28);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(20,24)(26,32)(28,29);(3,8)(5,7)(17,20,21,24)(18,25,23,19)(22,28,30,29)(26,27,32,31)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6607647);