Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | #33516 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
583 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 583 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,4,6,5,3,8,7)(9,10,14,22,15,23,35,21,17,25,34,40,16,24,13,20)(11,26,28,38,29,39,27,37,31,19,33,18,30,36,12,32);(1,3)(2,5)(4,6)(7,8)(9,11,15,30,17,31,16,29)(10,18,23,37,25,38,24,32)(12,13,33,14,27,35,28,34)(19,20,39,22,26,21,36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,19)(11,13)(14,30)(15,33)(16,28)(17,27)(18,20)(21,38)(22,37)(23,39)(24,36)(25,26)(29,34)(31,35)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44574);