Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,21,12)(8,22,24,31)(9,14)(10,16)(11,15)(13,25,32,26)(18,30)(19,29,20,28)(23,33)(27,35)(34,36);(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,29)(15,17)(16,28)(18,26)(22,33)(24,35)(25,34)(27,31)(32,36);(2,4)(5,8)(6,13,15,32)(7,18,21,34)(9,25)(10,26)(11,24)(12,27)(14,22,16,31)(17,33)(19,35,20,23)(28,36)(29,30);(2,4)(6,14)(9,10)(12,28)(13,22)(15,16)(17,29)(19,20)(23,35)(25,26)(27,36)(30,33)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295566);