#94578

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#99
commutator, prime radical	C₂³	C₂×C₃₂
Frattini, cosocle	C₂×M₆₄	C₂²
socle	C₂	#7362
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

75 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=ci,d^b=de,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, 1774);