C₁₂₇×#13545 #44577218

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13545

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2101
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

216 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

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, 6533);