Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 52 points:
(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(21,22,26,30)(23,31,32,49)(24,35,36,50)(25,40,38,29)(27,42,43,33)(28,45,46,39)(34,51,41,47)(37,48,52,44);(1,2)(3,4)(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)(24,36)(28,46)(31,49)(33,42)(35,50)(37,52)(39,45)(47,51);(2,4)(21,23,26,32)(22,27,30,43)(24,37,36,52)(25,41,38,34)(28,47,46,51)(29,48,40,44)(31,50,49,35)(33,45,42,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,d^c=di,e^c=eh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50986);