C₁₂₇×#8699 #44572372

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#8699

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄²×C₁₂₇ ◁ G

598 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 16256

A minimal faithful permutation representation — 3 generators on 159 points:

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

Conjugacy classes & character table

8128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 1687);