Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479394 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17204 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 817 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,12,6,24)(3,13,15,27,16,26,5,9)(4,17,20,30,22,29,21,10)(14,25,18,28,19,31,23,32);(1,3,6,5,8,16,7,15)(2,9,11,13,12,27,24,26)(4,18,20,19,22,23,21,14)(10,28,29,25,30,32,17,31);(1,4,6,21,8,22,7,20)(2,10,11,17,12,30,24,29)(3,14,15,18,16,19,5,23)(9,25,26,32,27,31,13,28);(1,5,7,16,8,15,6,3)(2,9,24,13,12,27,11,26)(4,19,20,18,22,14,21,23)(10,28,29,31,30,32,17,25)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386591);