C₁₂₇×#7522 #44571195

order	32512
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7522

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₁₂₇
Frattini, cosocle	#494	C₂²×C₁₂₇
prime radical	C₂²×C₁₂₇	AΣL(1,𝔽₁₆)
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ #494×C₁₂₇ ◁ G

142 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 30 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:

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

Conjugacy classes & character table

2794 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 510);