C₁₂₇×#12893 #44576566

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₂₇×#12893

Subgroups

	subgroup	G/H
center	C₄×C₁₂₇	#427
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1835
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ #117×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 191 points:

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

Conjugacy classes & character table

6604 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5881);