Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126351 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 288 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,16,21,15,23,14,22)(11,24,32,34,28,38,27,33)(12,29,31,18,30,20,13,19)(17,26,35,25,37,40,36,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,28)(10,17)(12,26,30,40)(13,25,31,39)(14,32,16,27)(18,33)(19,34)(20,38)(21,36)(22,35)(23,37)(24,29);(3,7)(5,8)(10,18)(12,13)(14,16)(17,33)(19,23)(20,22)(21,29)(24,36)(25,26)(27,32)(30,31)(34,37)(35,38)(39,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72265439, #72265452, #72265461, … |
| maximal normal subgroup of | #72265439, #72265452, #72265461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33693);