#94126

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#99
commutator	C₂³	C₄×C₁₆
Frattini, cosocle	C₂×#504	C₂²
socle	C₂	#7199
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

167 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,f^a=fi,d^b=df,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1322);