Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| 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 | #59124 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26);(5,7,9,6,11,8,10,12)(15,21)(16,24)(18,25)(19,23);(2,4)(6,12)(7,8)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(5,8,11,7)(6,10,12,9)(15,21)(18,25)(22,27)(26,28);(5,9,11,10)(6,7,12,8)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074173);