Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1404 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)(7,16,20,19)(8,21,22,23)(12,17,26,30)(13,27,29,28);(1,2)(3,4)(5,6)(7,17)(8,13)(9,14)(10,24)(11,15)(12,19)(16,26)(18,32)(20,30)(21,27)(22,29)(23,28)(25,33)(31,36)(34,35);(5,7)(6,12)(8,18)(9,19)(10,16)(11,20)(13,25)(14,30)(15,26)(17,24)(21,31)(22,34)(23,33)(27,32)(28,35)(29,36);(1,3,4,2)(5,8)(6,13)(7,18)(9,23)(10,21)(11,22)(12,25)(14,28)(15,29)(16,31)(17,32)(19,33)(20,34)(24,27)(26,36)(30,35)Polycyclic presentation:
2^9|a^2=h,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290664);