#94611

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₃₂
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	#7383
prime radical	#2262	C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

74 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=eg,d^2=f,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ce,e^b=eg,g^b=gh,h^b=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1807);