C₁₂₇×#12721 #44576394

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

Aliases

direct product

C₁₂₇×#12721

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×#41	C₂³×C₁₂₇
prime radical	#535×C₁₂₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄×#41×C₁₂₇ ◁ G

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

normal subgroups only

This group has 694 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)(3,12,15,27)(4,9,17,25)(5,10)(6,11)(7,13)(8,16,24,31)(14,20,18,32)(19,23)(21,28)(22,29)(26,30)(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)(4,13)(5,18)(6,15)(8,20)(9,12)(10,26)(11,22)(14,19)(16,30)(17,29)(21,23)(24,32)(25,27)(28,31);(1,4,6,17)(2,8)(3,13,15,29)(5,16,19,31)(7,20)(9,10)(11,24)(12,26)(14,28,18,30)(21,27)(22,32)(23,25)

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, 5709);