C₁₂₇×#10080 #44573753

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

Aliases

direct product

C₁₂₇×#10080

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
prime radical	C₂×#127×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

552 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 155
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 155 points:

(6,11)(7,12)(8,9)(14,18)(16,23)(19,27)(20,24)(21,25)(26,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14,17,20)(15,19,22,28)(16,24,23,18)(21,26,25,27);(2,4)(5,7,10,6)(8,12,9,11)(13,15,17,22)(14,19,20,28)(16,25,23,21)(18,26,24,27)

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, 3068);