C₁₂₇×#7338 #44571011

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#7338

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1316
commutator	C₂⁴	C₂×C₈×C₁₂₇
Frattini, cosocle	#379	C₂²×C₁₂₇
prime radical	C₂³×D₄×C₁₂₇	C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂⁴×C₁₂₇ ◁ #379×C₁₂₇ ◁ G

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

normal subgroups only

This group has 490 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) = 143
transitive degree	μ_t(G) = 2032

A minimal faithful permutation representation — 2 generators on 143 points:

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

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, 326);