C₁₂₇×#12313 #44575986

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

Aliases

direct product

C₁₂₇×#12313

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₁₂₇×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ G

356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

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, 5301);