C₁₂₇×#13564 #44577237

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#13564

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	#124	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₂×AΣL(1,𝔽₁₆)
prime radical	C₁₂₇×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #532×C₁₂₇ ◁ G

494 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 143
transitive degree	μ_t(G) = 2032

A minimal faithful permutation representation — 3 generators on 143 points:

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

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, 6552);