C₁₂₇×#7429 #44571102

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

Aliases

direct product

C₁₂₇×#7429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#354
commutator	C₂×C₄²	C₂×C₄×C₁₂₇
Frattini, cosocle	#391	C₂²×C₁₂₇
prime radical	C₂³×C₁₂₇	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #391×C₁₂₇ ◁ G

196 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 167
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 167 points:

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

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 417);