C₁₂₇×#11870 #44575543

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

Aliases

direct product

C₁₂₇×#11870

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1519
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
prime radical	C₁₂₇×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

392 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 4858);