Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136885 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(2,5,6,8)(9,10,14,19,15,20,27,37,16,21,13,29,12,18,28,35)(11,22,25,33,23,36,31,34,26,39,24,38,32,40,30,17);(3,7)(5,8)(9,11)(10,17,21,34)(12,23)(13,30)(14,31)(15,32)(16,26)(18,33,20,38)(19,36,29,40)(22,35,39,37)(24,28)(25,27);(1,2,4,6)(3,8,7,5)(9,12,16,15)(10,18,21,20)(11,23,26,32)(13,27,14,28)(17,33,34,38)(19,35,29,37)(22,36,39,40)(24,30,25,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=hi,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309476, #72309481, #72309482, … |
| maximal normal subgroup of | #72309476, #72309481, #72309482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44115);