C₁₂₇×#19618 #44583291

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₂₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₂₇×#19618

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#127
prime radical	C₂³×C₁₂₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

690 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 278 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) = 16256

A minimal faithful permutation representation — 4 generators on 159 points:

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

Conjugacy classes & character table

6604 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 12606);