C₁₂₇×#13700 #44577373

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13700

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ #131×C₁₂₇ ◁ #504×C₁₂₇ ◁ G

188 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Conjugacy classes & character table

5842 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6688);