Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10267 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 365 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12)(3,13)(4,20,23,10)(5,9,22,30)(6,25,24,11)(8,14)(15,29)(16,27)(17,26)(18,31)(19,32)(21,28);(1,3)(2,8)(4,21)(5,14)(6,18)(7,19)(9,26)(10,13)(11,28)(12,29)(15,22)(16,23)(17,24)(20,32)(25,27)(30,31);(2,9)(3,14)(4,22)(5,23)(6,24)(10,11)(12,30)(13,26)(15,19)(16,17)(18,21)(20,25)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fg,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51253);