C₁₂₇×#13513 #44577186

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13513

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#2085
prime radical	C₂³×C₁₂₇	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

228 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 191
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 191 points:

(1,2,5,23,6,12,8,11)(3,14,16,9,18,34,30,35)(4,10,20,39,22,40,31,37)(7,26,24,41,25,13,29,57)(15,46,47,62,48,44,49,32)(17,42,27,61,51,45,58,33)(19,52,53,36,54,64,50,63)(21,55,28,60,56,38,59,43)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,5,16,6,18,8,30)(2,9,11,14,12,35,23,34)(4,19,20,53,22,54,31,50)(7,27,24,51,25,58,29,17)(10,36,37,52,40,63,39,64)(13,42,41,33,26,45,57,61)(15,28,47,56,48,59,49,21)(32,60,44,55,62,43,46,38);(1,4,5,20,6,22,8,31)(2,10,11,37,12,40,23,39)(3,15,16,47,18,48,30,49)(7,28,24,56,25,59,29,21)(9,32,14,44,35,62,34,46)(13,43,41,38,26,60,57,55)(17,50,27,19,51,53,58,54)(33,52,45,63,61,64,42,36)

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6501);