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 | #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)(5,6,10,14,11,16,9,15)(7,17,20,12,21,28,19,29)(8,22,25,31,26,32,24,13)(18,27,34,33,35,36,23,30);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,27)(24,35)(26,34)(31,33)(32,36);(2,4)(5,8)(6,13,16,31)(7,18,21,35)(9,25)(10,24)(11,26)(12,27)(14,32,15,22)(17,33)(19,34,20,23)(28,30)(29,36);(2,4)(7,19)(9,10)(12,28)(14,15)(17,29)(18,34)(20,21)(22,32)(23,35)(24,25)(27,30)(33,36)Polycyclic presentation:
2^9|a^2=h,c^2=eh,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, 295588);