Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ 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:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,21)(17,24)(18,23)(20,22)(25,27)(26,28);(5,7,9,8)(6,10,11,12)(13,15,19,14)(16,25,18,28)(17,24,20,22)(21,26,23,27);(1,2,3,4)(13,16,19,18)(14,21,15,23)(17,26,20,27)(22,25,24,28);(5,8,9,7)(6,10,11,12)(13,17)(14,22)(15,24)(16,18)(19,20)(25,28);(13,18,19,16)(14,23,15,21)(17,27,20,26)(22,28,24,25)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491106);