Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100227 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| 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 | |
1168 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)(7,13,15,10)(8,11)(9,12)(14,17,20,18)(16,19)(21,22)(23,31,34,28)(24,27)(25,29)(26,30)(32,36,33,35);(2,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,20)(17,19)(22,27)(23,32)(24,25)(29,30)(33,34)(35,36);(5,8,9,16)(6,11,12,19)(7,14,15,20)(10,17,13,18)(21,23,26,34)(22,28,30,31)(24,32,25,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22193);