Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(26,29)(27,30)(31,32);(1,2,4,5)(3,6,7,8)(17,19)(18,21)(20,23)(22,24);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,23)(21,24)(25,26,28,29)(27,30,31,32);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=ei,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369593);