C₁₂₇×#13056 #44576729

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₂₇×#13056

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₂₇
prime radical	C₂×2₋¹⁺⁴×C₁₂₇	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×Q₈×C₁₂₇ ◁ G

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 8128

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

(2,4)(5,6,8,16,11,17,9,14)(7,18,21,12,22,29,19,30)(10,15,24,13,26,31,23,32)(20,33,34,36,25,27,35,28)(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)(5,7)(6,12)(8,19)(9,21)(10,25)(11,22)(13,27)(14,29)(15,28)(16,18)(17,30)(20,26)(23,35)(24,34)(31,36)(32,33);(5,8,11,9)(6,13,17,32)(7,19,22,21)(10,24,26,23)(12,27,30,33)(14,31,16,15)(18,28,29,36)(20,35,25,34)

Conjugacy classes & character table

4318 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6044);