Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 | #59112 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1975 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)(3,4)(5,6,9,11,10,12,7,8)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(6,8)(7,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(2,4)(5,7,10,9)(6,11,12,8)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27);(5,8,9,6,10,11,7,12)(16,24)(19,28)(21,23)(25,27);(5,7,10,9)(6,11,12,8)(14,20)(15,22)(19,28)(21,23)Polycyclic presentation:
2^9|a^2=fhi,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011952);