C₁₂₇×#12708 #44576381

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₂₇×#12708

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1781
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×D₄	C₂³×C₁₂₇
prime radical	C₁₂₇×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×#41×C₁₂₇ ◁ G

1002 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,11,14,25)(4,8)(5,17,20,9)(6,10)(7,13)(12,28,30,27)(15,26,19,23)(16,24)(18,22)(21,29)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,25)(10,22)(11,17)(13,20)(15,31)(16,30)(19,29)(23,27)(24,32)(26,28);(1,4,6,16)(2,8,10,24)(3,12,14,30)(5,19,20,15)(7,21,22,32)(9,26,17,23)(11,27,25,28)(13,31,18,29)

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, 5696);