Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #447 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10,14,18,15,20,12,16)(11,21,24,35,25,38,22,37)(13,27,30,19,28,34,26,36)(17,29,31,23,33,40,32,39);(2,6)(3,4)(7,8)(10,16,20,18)(13,28)(17,31,33,32)(19,34,36,27)(21,37,38,35)(23,39)(26,30)(29,40);(2,6)(3,4)(7,8)(9,11,15,25)(10,17,20,33)(12,24,14,22)(13,29,28,40)(16,31,18,32)(19,35,36,37)(21,27,38,34)(23,30,39,26)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=gi,e^2=gh,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309598, #72309605, #72309613, … |
| maximal normal subgroup of | #72309598, #72309605, #72309613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44715);