Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,19)(8,13,9,12)(11,17,20,16)(21,22,24,23)(25,27,33,30)(26,32,34,29)(28,36,31,35);(2,4)(5,7,10,18)(6,11,14,20)(8,17,9,16)(12,19,13,15)(21,23)(22,24)(25,30)(26,29)(27,33)(28,36)(31,35)(32,34);(7,16)(8,9)(11,15)(12,13)(17,18)(19,20)(22,27)(23,30)(25,35)(26,34)(28,29)(31,32)(33,36)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,h^b=hi,d^c=dg,e^c=eh,f^c=fgh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51353);