C₁₂₇×#8308 #44571981

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₂₇×#8308

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂⁴×C₄×C₁₂₇ ◁ G

1174 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 155
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 155 points:

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

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, 1296);