C₁₂₇×#10495 #44574168

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#10495

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

344 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 159 points:

(1,2)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18,22,26)(19,27,28,31)(20,23,21,25)(24,29,32,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27)

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 3483);