Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,7,5,3,4,8)(9,10,14,19,15,21,32,39,16,22,12,29,31,34,13,20)(11,23,26,36,27,35,25,37,28,40,24,18,33,38,30,17);(1,3,5,2)(4,8,6,7)(9,11)(10,17,22,37)(12,30)(13,24)(14,25)(15,33)(16,28)(18,34,36,21)(19,35,29,38)(20,40,39,23)(26,32)(27,31);(1,4,5,6)(2,7,3,8)(9,11)(10,18)(12,30)(13,24)(14,25)(15,33)(16,28)(17,34)(19,23)(20,35)(21,37)(22,36)(26,32)(27,31)(29,40)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ghi,d^2=g,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44132);