Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,10)(11,12)(21,22)(23,27)(24,28)(25,29)(26,30)(31,36)(32,34)(33,35);(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,19,20,17)(21,23)(22,27)(24,32)(25,33)(26,31)(28,34)(29,35)(30,36);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19)(22,28)(24,25)(27,34)(29,30)(32,33)(35,36);(5,8,9,7)(6,11,10,12)(22,29)(23,31)(24,25)(27,34)(28,30)(35,36)Polycyclic presentation:
2^9|b^2=g,c^2=eg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353702);