C₁₂₇×#7227 #44570900

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#7227

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₂₇
Frattini, cosocle	C₂×#131	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1289
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×C₈×C₁₂₇ ◁ G

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

normal subgroups only

This group has 248 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) = 191
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 191 points:

(1,2)(3,13)(4,18,20,44)(5,23)(6,11)(7,12)(8,14,26,38)(9,22,29,46)(10,31)(15,41)(16,35)(17,36)(19,32)(21,48,47,30)(24,33)(25,51)(27,54,52,62)(28,40,53,59)(34,55)(37,57)(39,61,60,50)(42,58)(43,63)(45,49,64,56)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,5,15,6,16,24,42,7,17,25,43,19,37,31,55)(2,8,10,27,32,53,51,61,12,26,33,52,11,28,23,50)(4,14,21,39,22,40,49,62,20,38,47,60,46,59,56,54)(9,13,30,34,44,57,64,63,29,36,48,58,18,35,45,41)

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 215);