C₁₂₇×#11841 #44575514

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

Aliases

direct product

C₁₂₇×#11841

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₂₇
prime radical	C₂⁵×C₁₂₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂⁴×C₁₂₇ ◁ C₂²×C₂²⋊C₄×C₁₂₇ ◁ G

506 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,11,13,25)(4,15,17,9)(5,8)(6,10)(7,12,20,30)(14,27,31,32)(16,21,29,22)(18,26,28,23)(19,24)(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)(2,7,10,21)(4,16,17,30)(5,13,19,31)(8,22,24,20)(9,25,15,32)(11,26,27,23)(12,28,29,18);(2,8)(4,17)(7,20)(9,26)(10,24)(11,25)(12,29)(15,23)(16,30)(18,28)(21,22)(27,32)

Conjugacy classes & character table

4699 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 4829);