Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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 | |
2206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14,17,22)(15,23,25,27)(16,21,18,20)(19,26,28,24);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,24)(25,28)(26,27);(2,4)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16,17,18)(14,20,22,21)(15,24,25,26)(19,23,28,27);(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,27,28,23)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346873);