Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,20,31)(8,12,23,30)(9,14)(10,15)(11,19,26,18)(13,24)(17,32)(21,35)(22,29,36,28)(25,33)(27,34);(1,2)(3,4)(5,7)(6,11)(8,21)(9,18)(10,20)(12,27)(13,16)(14,31)(15,26)(17,23)(19,24)(22,34)(25,30)(28,35)(29,32)(33,36);(2,4)(5,8)(6,12)(7,17)(9,22)(10,23)(11,25)(13,28)(14,29)(15,30)(16,32)(18,33)(19,34)(20,21)(24,36)(26,27)(31,35);(6,13)(7,18)(9,24)(12,28)(14,15)(16,31)(17,33)(19,20)(21,34)(22,36)(29,30)(32,35)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290318);