C₁₂₇×#13689 #44577362

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13689

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
prime radical	C₂×D₃₂×C₁₂₇	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

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

normal subgroups only

This group has 216 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) = 163
transitive degree	μ_t(G) = 16256

A minimal faithful permutation representation — 3 generators on 163 points:

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

Conjugacy classes & character table

6223 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6677);