C₁₂₇×#7075 #44570748

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₂₇×#7075

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₂×C₈	C₂×C₈×C₁₂₇
Frattini, cosocle	C₈²	C₂²×C₁₂₇
socle, prime radical	C₂²×C₁₂₇	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₈²×C₁₂₇ ◁ G

134 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 32512

A minimal faithful permutation representation — 2 generators on 159 points:

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

Conjugacy classes & character table

7366 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 63);