Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,3)(5,6)(7,10)(8,11)(9,12)(13,14,20,26)(15,21,32,39)(16,22,34,42)(17,23,33,41)(18,25,19,24)(27,37,28,44)(29,40,43,36)(30,35,31,38);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,20,32)(14,21,26,39)(16,29,34,43)(17,27,33,28)(18,31,19,30)(22,40,42,36)(23,37,41,44)(24,35,25,38);(1,3,4,2)(6,10)(7,8)(11,12)(14,22)(15,27)(16,17)(18,19)(21,35)(23,24)(25,41)(26,42)(28,32)(29,31)(30,43)(33,34)(37,44)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=d,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=eg,f^c=fgi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52087);