C₁₂₇×#12095 #44575768

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

Aliases

direct product

C₁₂₇×#12095

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₂₇
prime radical	#534×C₁₂₇	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂×C₂²⋊C₄×C₁₂₇ ◁ G

458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 167
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 167 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,36)(22,35)(23,34)(24,30)(25,31)(26,33)(27,37)(28,38)(29,39)(32,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,28)(12,23,25,20)(13,21,26,38)(16,29,31,27)(17,19,33,35)(24,39,40,37)(30,34,32,36);(9,12,14,25)(10,16,18,31)(11,20,22,23)(13,24,26,40)(15,27,28,29)(17,32,33,30)(19,34,35,36)(21,37,38,39)

Conjugacy classes & character table

3175 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5083);