Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142969 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 679 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(23,28)(25,32);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(4,14)(6,15)(7,12)(9,16)(17,19,21,26)(18,22,24,31)(20,29,30,27)(23,32,28,25)Polycyclic presentation:
2^9|a^2=g,c^2=gi,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 | #72235698, #72235719, #72235808, … |
| maximal normal subgroup of | #72235698, #72235719, #72235808, … |
| maximal quotient of | #72237857, #72237861, #72285022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50195);