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 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,26)(19,23,22,25)(20,28,29,24)(27,32,30,31);(1,2)(3,4)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,31,28,32);(2,4)(5,6,8,7)(10,14)(13,15)(17,20,21,29)(18,24,26,28)(19,27,22,30)(23,31,25,32);(5,7,8,6)(11,16)(13,15)(18,25)(19,22)(23,26)(24,32)(27,30)(28,31)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,d^2=efi,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, 338625);