C₁₂₇×#7229 #44570902

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#7229

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₂₇
Frattini, cosocle	C₂×#131	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1289
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₈×C₁₂₇ ◁ G

244 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

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, 217);