Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128009 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
796 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 796 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 3 generators on 32 points:
(2,8,10,24)(3,14)(4,16)(5,20)(9,27,28,30)(11,25,29,23)(12,26,13,17)(15,18)(19,32)(21,31);(1,2,5,11,6,12,3,9)(4,17,18,30,16,24,31,23)(7,13,20,28,22,10,14,29)(8,21,25,32,26,15,27,19);(1,3,6,5)(2,9,12,11)(4,15,16,21)(7,14,22,20)(8,23,26,30)(10,28,13,29)(17,27,24,25)(18,19,31,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,d^b=dh,e^b=egi,h^b=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233221, #72233226, #72233228, … |
| maximal normal subgroup of | #72233221, #72233226, #72233228, … |
| maximal quotient of | C2×#128179 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35375);