C₁₂₇×#12969 #44576642

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

Aliases

direct product

C₁₂₇×#12969

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

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

normal subgroups only

This group has 416 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:

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

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5957);