C₁₂₇×#11770 #44575443

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₂₇×#11770

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1505
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₂₇
prime radical	C₁₂₇×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,8)(4,17,19,7)(5,10)(6,12)(9,18,21,28)(11,14)(13,23)(15,29,30,20)(16,25)(22,26,31,32)(24,27)(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);(2,7)(3,13)(4,18)(5,14)(8,22)(9,11)(10,21)(12,17)(15,30)(16,27)(19,28)(20,23)(24,29)(25,31);(1,3,14,27,6,16,5,13)(2,8,11,24,12,25,10,23)(4,15,28,32,19,30,18,26)(7,20,21,31,17,29,9,22)

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, 4758);