C₁₂₇×#12255 #44575928

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

Aliases

direct product

C₁₂₇×#12255

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂×C₂²⋊C₄×C₁₂₇ ◁ G

378 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 8128

A minimal faithful permutation representation — 3 generators on 159 points:

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

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, 5243);