C₁₂₇×#12047 #44575720

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#12047

Subgroups

	subgroup	G/H
center	C₈×C₁₂₇	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₈×D₈
prime radical	#508×C₁₂₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂×C₁₆×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Conjugacy classes & character table

8128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5035);