C₂×C₁₂₇×#1711 #44583272

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₂₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×C₁₂₇×#1711

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂×#127×C₁₂₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

802 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 161
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 4 generators on 161 points:

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

Conjugacy classes & character table

9652 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 12587);