C₁₂₇×#7207 #44570880

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#7207

Subgroups

	subgroup	G/H
center	C₈×C₁₂₇	#99
commutator	C₂³	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×#131	C₂²×C₁₂₇
socle	C₂×C₁₂₇	#1282
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂×C₈×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

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, 195);