Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
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 | |
289 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14)(15,26)(16,20)(17,33)(18,35)(19,25)(21,28)(22,31)(23,34)(24,32)(27,39)(29,36)(30,37)(38,43)(40,42)(41,44);(5,7,9,6)(8,11,12,10)(14,20)(15,27)(16,31)(17,34)(18,32)(22,24)(23,35)(25,33)(26,40)(28,30)(29,44)(36,37)(38,41)(42,43);(2,4)(6,10)(7,11)(8,12)(13,15,16,27,32,42,34,44,19,30,17,28,18,29,31,43)(14,21,22,37,35,41,33,40,25,39,23,26,24,38,20,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egh,f^c=fgh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52776);