C₁₂₇×#7216 #44570889

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₂₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₂₇×#7216

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#342
commutator	C₂²×C₄	C₄²×C₁₂₇
Frattini, cosocle	C₂×#117	C₂²×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ C₂×#117×C₁₂₇ ◁ G

276 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 167
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 2 generators on 167 points:

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

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, 204);