Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| 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 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,31)(21,24)(22,27)(23,26)(25,29)(30,32);(5,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,32)(26,27)(28,31);(6,7)(11,15)(13,16)(17,19,22,29,21,20,23,30)(18,25,26,28,24,32,27,31);(2,4)(5,7,8,6)(17,20,23,29,21,19,22,30)(18,25,27,31,24,32,26,28)Polycyclic presentation:
2^9|a^2=egi,b^2=ei,c^2=f,d^2=ef,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^c=de80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370523);