Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142966 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #18164 | C2 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 667 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:
(1,2,7,10)(3,11,15,29)(4,16,20,19)(5,6,21,22)(8,23,24,13)(9,25,27,18)(12,28,30,17)(14,26,32,31);(1,3)(2,8)(4,17)(5,14)(6,12)(7,15)(9,26)(10,24)(11,16)(13,18)(19,29)(20,28)(21,32)(22,30)(23,25)(27,31);(2,6)(3,12)(8,14)(9,16)(10,22)(11,28)(13,31)(15,30)(17,29)(19,27)(23,26)(24,32)Polycyclic presentation:
2^9|a^2=i,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72262526, #72262740, #72262743, … |
| maximal normal subgroup of | #72262526, #72262740, #72262743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50163);