C₁₂₇×#13718 #44577391

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13718

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×M₃₂×C₁₂₇ ◁ G

160 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,13,14,42)(4,18,19,52)(5,24,27,60)(6,28,26,11)(8,30,31,15)(9,34,35,22)(10,38,40,21)(16,50,46,64)(17,51,29,45)(20,56,48,39)(23,43,55,54)(25,44,61,36)(32,57,63,59)(33,49,41,62)(37,58,53,47)(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);(4,19)(9,35)(13,42)(15,30)(18,52)(20,48)(22,34)(23,55)(36,44)(37,53)(43,54)(45,51)(47,58)(49,62)(50,64)(57,59);(1,3,6,29,7,14,26,17)(2,8,11,41,12,31,28,33)(4,20,23,59,19,48,55,57)(5,25,21,16,27,61,38,46)(9,36,37,50,35,44,53,64)(10,39,24,32,40,56,60,63)(13,43,51,18,42,54,45,52)(15,47,62,22,30,58,49,34)

Conjugacy classes & character table

3937 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6706);