C₁₂₇×#12274 #44575947

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#12274

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₂₇
prime radical	C₁₂₇×#2828	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂×C₄⋊C₄×C₁₂₇ ◁ G

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,15,19,32)(4,8,20,10)(5,23,25,31)(7,14,26,13)(9,22,28,18)(11,21,30,17)(16,29,24,27)(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);(2,8)(3,16)(4,20)(5,21)(9,13)(10,12)(11,29)(14,28)(15,23)(17,25)(18,22)(19,24)(27,30)(31,32);(1,3)(2,9,14,10)(4,21)(5,18)(6,19)(7,16)(8,12,28,13)(11,15,23,29)(17,20)(22,25)(24,26)(27,30,32,31)

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, 5262);