Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
1462 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22,25,30,27,31,26,29)(23,32,33,36,24,35,34,28);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,27,24)(22,28,31,36)(25,34,26,33)(29,32,30,35);(2,4)(15,19)(17,20)(21,24,27,23)(22,28,31,36)(25,33,26,34)(29,32,30,35);(5,8,9,7)(6,10,11,12)(22,29)(25,26)(28,32)(30,31)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=eg,b^2=efi,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 310220);