C₁₂₇×#13076 #44576749

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#13076

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1893
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₁₂₇×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

466 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

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, 6064);