C₁₂₇×#13569 #44577242

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#13569

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	#124	C₂³×C₁₂₇
prime radical	C₂×#454×C₁₂₇	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×#124×C₁₂₇ ◁ G

492 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,11,14,29)(4,16,18,28)(5,20,22,9)(6,10)(7,13,24,31)(8,19)(12,26,15,30)(17,25,32,23)(21,27)(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)(2,7)(4,17)(5,14)(6,12)(8,26)(9,24)(10,23)(11,27)(13,18)(15,22)(16,30)(19,31)(20,25)(21,32)(28,29);(3,12)(7,23)(11,26)(13,17)(14,15)(24,25)(29,30)(31,32)

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6557);