Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100027 |
| 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 | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,16)(7,17,18,20)(8,12,19,13)(9,15,10,14)(21,22)(23,26)(24,31,32,27)(25,28)(29,35,36,33)(30,34);(1,2)(3,4)(5,7)(6,12)(8,10)(9,19)(11,18)(13,16)(14,20)(15,17)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(2,4)(5,8,9,18,11,19,10,7)(6,13,14,20,16,12,15,17)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26727);