C₁₂₇×#13594 #44577267

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

Aliases

direct product

C₁₂₇×#13594

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#354
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	#127	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₂×#354
prime radical	#532×C₁₂₇	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #532×C₁₂₇ ◁ G

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

normal subgroups only

This group has 326 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)(3,12,15,22)(4,8)(5,10,19,9)(6,11)(7,14,21,18)(13,26,28,32)(16,24,30,23)(17,25)(20,27,31,29)(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);(1,3,6,14)(2,7)(4,13,17,27)(5,18,19,15)(8,20)(9,12)(10,21)(11,22)(16,29,30,28)(23,26)(24,31)(25,32);(1,4,6,17)(2,8,11,25)(3,13,14,27)(5,16,19,30)(7,20,22,32)(9,23,10,24)(12,26,21,31)(15,28,18,29)

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, 6582);