C₁₂₇×#11522 #44575195

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

Aliases

direct product

C₁₂₇×#11522

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#410
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1464
prime radical	C₂⁴×C₁₂₇	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×M₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 284 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:

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

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, 4510);