C₁₂₇×#19080 #44582753

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₂₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₂₇×#19080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₁₂₇×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

936 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 16256

A minimal faithful permutation representation — 4 generators on 159 points:

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

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, 12068);