Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #466 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,12,9,8,10,11)(13,14,20,21)(15,26,23,16)(17,27,28,22)(18,25,19,24);(1,2)(3,4)(5,7,9,10)(6,11,8,12)(13,15,17,25,18,27,16,14,20,23,28,24,19,22,26,21);(2,4)(5,8,9,6)(7,12,10,11)(13,16)(14,21)(15,27)(17,18)(19,28)(20,26)(22,23)Polycyclic presentation:
2^9|a^2=de,b^2=dgi,c^2=gh,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53142);