Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1726 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)(7,8)(9,10,13,16)(11,14,12,15)(17,18)(19,25)(20,30)(21,23)(22,28)(24,26)(27,31)(29,32);(1,2)(3,4)(9,11,13,12)(10,14,16,15)(17,19,26,27,21,29,20,28)(18,22,30,32,23,31,24,25);(2,4)(9,11,13,12)(10,15,16,14)(18,23)(22,31)(24,30)(25,32);(6,8)(9,12,13,11)(10,15,16,14)(19,27)(20,26)(22,32)(24,30)(25,31)(28,29)Polycyclic presentation:
2^9|a^2=fi,b^2=ef,c^2=fi,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339327);