Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141213 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #469 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 96 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)(9,10,15,22,17,23,16,35)(11,24,26,40,27,36,28,19)(12,18,29,37,25,39,34,38)(13,30,31,21,14,33,32,20);(2,6)(3,4)(7,8)(9,11,13,12,16,26,32,29,17,27,14,25,15,28,31,34)(10,18,21,24,22,38,33,19,23,39,20,36,35,37,30,40);(9,12,14,26,16,29,31,27,17,25,13,28,15,34,32,11)(10,19,21,18,22,36,33,38,23,40,20,39,35,24,30,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72309606, #72309613, #72309614, … |
| maximal normal subgroup of | #72309606, #72309613, #72309614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48476);