C₁₂₇×#10669 #44574342

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#10669

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#399
prime radical	C₂³×C₁₂₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 8128

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

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

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 3657);