C₁₂₇×#12450 #44576123

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#12450

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₈²×C₁₂₇ ◁ G

206 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 32512

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

(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18)(19,27)(20,23)(21,33)(22,36)(24,29)(25,35)(26,32)(28,43)(30,47)(31,44)(34,42)(37,41)(38,45)(39,46)(40,48)(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);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,23)(24,38)(25,39)(26,36)(27,43)(33,44)(40,42)(41,47);(17,19,20,28)(18,24,23,38)(21,34,31,48)(22,37,32,30)(25,41,39,47)(26,42,36,40)(27,44,43,33)(29,46,45,35)

Conjugacy classes & character table

5461 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5438);