Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143939 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19187 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(2,6)(3,4)(7,8)(9,10,14,19)(11,15,23,32)(12,16,27,35)(13,28,26,18)(17,36,34,25)(20,30,22,31)(21,39,24,33)(29,40,37,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,24)(13,20)(14,23)(16,33)(17,37)(18,31)(19,32)(21,27)(22,26)(25,40)(28,30)(29,34)(35,39)(36,38);(2,6)(3,4)(7,8)(10,16)(11,20)(12,25)(13,26)(15,29)(17,18)(19,35)(21,24)(22,23)(27,36)(28,34)(30,39)(31,33)(32,37)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,d^c=dg,e^c=eh,f^c=fgi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51136);