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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2203 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)(15,22)(16,20)(17,21)(18,19)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,17,18)(14,19,21,22)(16,24,25,23)(20,27,28,26);(2,4)(5,8,9,7)(6,10,11,12)(14,19)(15,18)(20,26)(21,22)(23,24)(27,28);(2,4)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,28)(22,27);(14,21)(16,25)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346410);