C₁₂₇×#8556 #44572229

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₂₇×#8556

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#379
commutator	C₂⁴	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

842 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 159 points:

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

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, 1544);