Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,31)(25,34)(26,29)(28,30)(32,36)(33,35);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,23,26,25)(22,27,29,34)(24,32,33,30)(28,31,36,35);(2,4)(5,8,9,6)(7,11,12,10)(14,18)(17,19);(7,12)(10,11)(15,20)(17,19)(21,24,26,33)(22,28,29,36)(23,30,25,32)(27,35,34,31)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332256);