Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123011 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
310 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,13)(10,15)(11,16)(12,14)(17,18,24,30)(19,31,20,34)(21,37,40,47)(22,29,23,28)(25,35,45,32)(26,33,46,36)(27,41,42,39)(38,43,48,44);(2,6)(7,13)(8,14)(15,16)(17,19,24,20)(18,25,30,45)(21,38,40,48)(22,33,23,36)(26,42,46,27)(28,44,29,43)(31,47,34,37)(32,41,35,39);(1,3,5,10)(2,7,8,15)(4,11,12,9)(6,13,14,16)(17,20,24,19)(18,26,30,46)(21,32,40,35)(22,33,23,36)(25,42,45,27)(28,31,29,34)(37,44,47,43)(38,41,48,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30224);