C₁₂₇×#7433 #44571106

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7433

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₂₇
Frattini, cosocle	#407	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1352
prime radical	C₂⁴×C₁₂₇	Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #407×C₁₂₇ ◁ G

182 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 159 points:

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

Conjugacy classes & character table

3937 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 421);