C₁₂₇×#12513 #44576186

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

Aliases

direct product

C₁₂₇×#12513

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₁₂₇×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₈²×C₁₂₇ ◁ G

290 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(18,25)(19,29)(20,21)(22,23)(26,28)(27,31)(30,32)(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);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,18,21,26,22,27,30,19,24,29,32,31,23,28,20,25);(2,7)(6,14)(8,12)(9,15)(17,19,30,28,23,26,21,29,24,25,20,27,22,31,32,18)

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, 5501);