Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143148 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18)(19,28)(20,24)(21,26)(22,23)(25,30)(27,31)(29,32);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19)(18,23)(20,31)(21,29)(22,30)(24,28)(25,32);(6,15)(9,16)(18,24)(19,29)(20,30)(21,31)(23,32)(25,27)(26,28)Polycyclic presentation:
2^9|d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#143148, #72213171, #72213174, … |
| maximal normal subgroup of | C2×#143148, #72213171, #72213174, … |
| maximal quotient of | #72213148, C2×#143148 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50344);