C₁₂₇×#12169 #44575842

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

Aliases

direct product

C₁₂₇×#12169

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

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

normal subgroups only

This group has 342 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) = 167
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 167 points:

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

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, 5157);