C₁₂₇×#7226 #44570899

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7226

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₂₇
Frattini, cosocle	C₄×M₁₆	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1288
prime radical	C₂³×C₁₂₇	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₈×C₁₂₇ ◁ G

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 159
transitive degree	μ_t(G) = 4064

A minimal faithful permutation representation — 2 generators on 159 points:

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

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 214);