C₁₂₇×#13432 #44577105

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

Aliases

direct product

C₁₂₇×#13432

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₁₂₇
prime radical	C₁₂₇×#3199	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#41×C₁₂₇ ◁ G

440 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 86 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)(4,7)(5,18)(6,11)(8,15)(9,16)(10,19)(13,26)(14,29)(17,24)(20,30)(21,31)(22,32)(23,28)(25,27)(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)(4,16)(8,20)(9,11)(10,21)(12,25)(17,31)(18,24)(22,23)(26,29)(30,32);(1,3,4,14,6,13,16,27)(2,8,9,22,11,23,7,20)(5,15,17,30,19,28,31,32)(10,12,24,25,18,26,21,29)

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, 6420);