C₁₂₇×#7438 #44571111

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7438

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₂₇
Frattini, cosocle	#408	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1352
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #408×C₁₂₇ ◁ G

174 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 191 points:

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

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, 426);