C₁₂₇×#21827 #44585500

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₂₇×#21827

Subgroups

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

1130 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 366 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 — 4 generators on 159 points:

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

Conjugacy classes & character table

5842 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 14815);