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 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 373 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) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 52 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,19)(8,13,9,12)(11,17,20,16)(21,22,26,30)(23,31,34,50)(24,35,39,49)(25,40,37,29)(27,42,44,32)(28,45,47,38)(33,51,41,46)(36,48,52,43);(2,4)(5,7,10,18)(6,11,14,20)(8,17,9,16)(12,19,13,15);(7,16)(8,9)(11,15)(12,13)(17,18)(19,20)(21,23,26,34)(22,27,30,44)(24,36,39,52)(25,41,37,33)(28,46,47,51)(29,48,40,43)(31,49,50,35)(32,45,42,38)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eh,f^c=fg,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51194);