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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,24,20,25)(21,31,28,26)(23,30,32,29);(1,3,4,2)(9,11,12,16)(10,13,14,15)(17,19,28,30,22,20,21,29)(18,23,31,25,27,32,26,24);(2,3)(17,20,22,19)(18,24,27,25)(21,29,28,30)(23,31,32,26);(6,8)(11,16)(13,15)(17,20,22,19)(18,25,27,24)(21,29,28,30)(23,26,32,31)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340022);