C₁₂₇×#7449 #44571122

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7449

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1361
commutator	C₂×C₁₆	C₂×C₄×C₁₂₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₂₇
prime radical	C₂²×C₁₂₇	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

126 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 50 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:

(1,2,5,11,28,14,30,40,8,15,29,13,7,31,6,12)(3,16,18,47,32,49,56,64,21,50,55,39,20,57,19,48)(4,22,24,10,38,41,51,43,27,59,62,44,26,42,25,58)(9,23,33,17,36,52,45,63,37,61,35,54,46,60,34,53)(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);(1,3,8,21)(2,9,15,37)(4,23,27,61)(5,19,29,56)(6,18,30,55)(7,32,28,20)(10,39,44,47)(11,34,13,45)(12,33,40,35)(14,46,31,36)(16,41,50,42)(17,51,54,25)(22,57,59,49)(24,53,62,63)(26,52,38,60)(43,64,58,48)

Conjugacy classes & character table

4699 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 437);