C₁₂₇×#10664 #44574337

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#10664

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#399
prime radical	C₂³×C₁₂₇	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

304 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 159 points:

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

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 3652);