Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11);(2,3)(5,7,9,8)(6,10,11,12)(14,18)(15,21)(16,23)(20,24)(22,25)(26,28);(1,2,4,3)(5,8,9,7)(6,10,11,12)(13,14)(15,19)(16,24)(17,20)(18,23)(21,26)(22,27)(25,28);(1,3,4,2)(13,15,17,22)(14,19,20,27)(16,25,23,21)(18,26,24,28);(14,20)(18,24)(19,27)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354307);