C₁₂₇×#13473 #44577146

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13473

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2067
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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:

(1,2,9,14,8,15,7,34)(3,16,36,51,21,39,20,57)(4,22,35,47,27,43,26,46)(5,28,33,13,6,30,17,12)(10,37,40,31,41,52,50,19)(11,29,44,24,45,32,48,25)(18,53,58,59,55,38,54,64)(23,60,62,63,61,49,56,42)(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);(1,3,17,52,7,20,6,31,8,21,33,37,9,36,5,19)(2,10,28,51,14,40,13,39,15,41,30,57,34,50,12,16)(4,23,29,18,26,56,25,54,27,61,32,55,35,62,24,58)(11,42,46,53,44,60,22,59,45,63,47,38,48,49,43,64);(1,4,8,27)(2,11,15,45)(3,18,21,55)(5,29,6,32)(7,35,9,26)(10,38,41,53)(12,46,13,47)(14,48,34,44)(16,49,39,60)(17,24,33,25)(19,56,31,62)(20,58,36,54)(22,28,43,30)(23,37,61,52)(40,59,50,64)(42,51,63,57)

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, 6461);