Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| 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 | |
565 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,12,16,31)(4,8)(5,21)(6,22)(7,11)(9,15,28,32)(10,18)(13,24,23,26)(14,29,17,27)(19,30)(20,25);(2,8)(3,13,16,23)(9,12,28,31)(10,21)(11,25)(14,15,17,32)(22,30)(24,27,26,29);(1,3)(2,9)(4,17)(5,13)(6,23)(7,16)(8,24)(10,29)(11,28)(12,22)(14,20)(15,19)(18,32)(21,31)(25,26)(27,30)Polycyclic presentation:
2^9|a^2=di,b^2=di,e^2=gi,f^2=g,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,d^c=dg,e^c=egi,f^c=fg,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72262577, #72262749, #72262750, … |
| maximal normal subgroup of | #72262577, #72262749, #72262750, … |
| maximal quotient of | #72339022 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51812);