Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,12,17,10,16)(7,18,20,8,21,23,13,24)(9,25,28,32,29,14,27,31)(19,30,26,22,35,33,34,36);(5,7)(6,8)(9,26)(10,13)(11,20)(12,21)(14,30)(15,23)(16,18)(17,24)(19,27)(22,31)(25,33)(28,35)(29,34)(32,36);(2,4)(5,8)(6,13,17,20)(7,15,21,16)(9,22)(10,23)(11,18)(12,24)(14,26,25,34)(19,32,35,31)(27,33)(28,30)(29,36);(2,4)(5,9,12,29)(6,14,17,25)(7,19,21,35)(8,22,24,36)(10,28,11,27)(13,26,20,34)(15,32,16,31)(18,33,23,30)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295589);