Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479834 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17272 |
| 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 | |
811 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 811 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,8)(4,16,17,27)(5,9,19,10)(6,20,21,12)(13,23)(14,25)(15,30)(18,22)(24,28)(26,31)(29,32);(1,3)(2,8,11,25)(4,13)(5,18)(6,15)(7,14)(9,26,10,22)(12,24,20,30)(16,32,27,23)(17,29)(19,31)(21,28);(1,4,6,19,7,17,21,5)(2,9,20,16,11,10,12,27)(3,13,28,18,14,29,15,31)(8,22,24,32,25,26,30,23);(1,5,6,17,7,19,21,4)(2,10,12,27,11,9,20,16)(3,13,15,18,14,29,28,31)(8,23,24,26,25,32,30,22)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251690, #72251696, … |
| maximal normal subgroup of | #72251676, #72251690, #72251696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387030);