Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35329 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11,17,26)(8,19,20,28)(9,15)(10,14)(12,21,29,34)(13,22)(16,31)(18,25,23,30)(24,35)(27,33)(32,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,23)(12,27)(13,25)(14,30)(15,26)(18,22)(19,31)(20,32)(21,35)(24,29)(28,36)(33,34);(5,8)(6,12,15,29)(7,16)(9,20)(10,21)(11,24,26,27)(13,28,14,19)(17,32)(18,33)(22,34)(23,35)(25,31,30,36);(1,3,4,2)(6,13,15,14)(7,17)(11,25,26,30)(12,28,29,19)(16,32)(18,23)(24,31,27,36)(33,35)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293265);