C₁₂₇×#12968 #44576641

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₂₇×#12968

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×#41	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1867
prime radical	#532×C₁₂₇	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#41×C₁₂₇ ◁ G

424 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5956);