C₁₂₇×#12046 #44575719

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

Aliases

direct product

C₁₂₇×#12046

Subgroups

	subgroup	G/H
center	C₁₆×C₁₂₇	D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₈×D₈
prime radical	#508×C₁₂₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₂₇ ◁ C₂×C₁₆×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

216 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Conjugacy classes & character table

11176 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5034);