C₁₂₇×#12297 #44575970

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

Aliases

direct product

C₁₂₇×#12297

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	#466×C₁₂₇	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #424×C₁₂₇ ◁ G

290 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 175
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 175 points:

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

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, 5285);