C₂×C₁₂₇×#1358 #44575592

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₁₂₇×#1358

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#331
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

390 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 8128

A minimal faithful permutation representation — 3 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);(3,4,6,11,7,10,19,26,8,13,20,29,22,12,30,28)(5,14,15,31,16,25,32,34,17,24,33,27,21,9,18,23);(3,5,8,17)(4,9,13,25)(6,18,20,32)(7,21,22,16)(10,24,12,14)(11,27,29,31)(15,19,33,30)(23,26,34,28)

Conjugacy classes & character table

5842 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 4907);