Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142969 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #18162 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,8,12)(3,13,16,30)(4,19,21,23)(5,10,7,6)(9,26,17,15)(11,22,29,24)(14,31,27,20)(18,28,25,32);(1,3)(2,9)(4,20)(5,25)(6,14)(7,18)(8,16)(10,27)(11,28)(12,17)(13,19)(15,24)(21,31)(22,26)(23,30)(29,32);(2,10)(3,14)(4,21)(6,12)(9,18)(11,19)(13,20)(15,28)(16,27)(17,25)(22,24)(23,29)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=i,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,h^b=hi,d^c=dh,f^c=fgi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72231834, #72233403, #72233495, … |
| maximal normal subgroup of | #72231834, #72233403, #72233495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50166);