C₁₂₇×#13431 #44577104

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

Aliases

direct product

C₁₂₇×#13431

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₁₂₇
prime radical	C₂×#41×C₁₂₇	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#41×C₁₂₇ ◁ G

402 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6419);