Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467093 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16558 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,5,11,6,20)(3,13,16,25,17,8,22,30)(4,9,21,27,18,10,19,28)(14,23,15,24,31,29,32,26);(1,3,6,16,5,17,7,22)(2,8,20,30,11,13,12,25)(4,15,19,31,18,32,21,14)(9,26,28,23,10,24,27,29);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,23)(10,11)(12,28)(13,29)(15,22)(16,32)(17,31)(20,27)(24,25)(26,30);(1,4)(2,10,11,9)(3,15)(5,18)(6,19)(7,21)(8,24,13,26)(12,28,20,27)(14,22)(16,31)(17,32)(23,30,29,25)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374289);