C₁₂₇×#11486 #44575159

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#11486

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	C₂×#99
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1459
prime radical	#532×C₁₂₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₈×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

364 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Conjugacy classes & character table

5842 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 4474);