C₁₂₇×#12729 #44576402

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₂₇×#12729

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₂₇
prime radical	#535×C₁₂₇	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ #519×C₁₂₇ ◁ G

690 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

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, 5717);