C₁₂₇×#13504 #44577177

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13504

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2077
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

278 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 159 points:

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

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, 6492);