Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,30)(24,27)(25,29)(26,32)(28,34)(31,36)(33,35);(5,7,9,14)(6,10,12,17)(8,13,15,20)(11,18,19,16)(21,23,25,33)(22,26,29,36)(24,31,34,32)(27,30,28,35);(5,8)(6,11)(7,13)(9,15)(10,16)(12,19)(14,20)(17,18)(22,27)(23,31)(24,34)(28,29)(30,35)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22447);