Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,28)(24,35)(25,31)(26,30)(27,32)(29,34)(33,36);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,27,24)(22,28,32,35)(25,33,26,34)(29,31,36,30);(2,4)(15,19)(17,20)(21,24,27,23)(22,29,32,36)(25,33,26,34)(28,31,35,30);(5,8,9,7)(6,10,11,12)(22,30)(25,26)(28,29)(31,32)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=eg,b^2=ei,c^2=i,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,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309128);