#94736

order	512
nilpotency class	6
isoclinism	isoclinic class of #94690
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
prime radical	C₂×D₃₂	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

85 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,6,4,5,7,8)(10,17,18,41)(11,23)(12,30)(13,25)(14,32)(15,38)(19,46,20,49)(21,51,45,57)(22,52,40,44)(24,31,42,35)(26,29)(27,62)(28,61)(33,39)(34,60)(36,67)(37,43,47,70)(48,63,50,72)(53,55)(54,59)(56,64)(58,69,66,68)(65,71);(2,5)(3,7)(9,10)(11,24)(12,31)(13,35)(14,37)(15,40)(16,18)(17,26)(19,30)(20,25)(21,32)(22,38)(23,41)(27,63)(28,66)(29,42)(33,45)(34,69)(36,68)(39,47)(43,71)(44,56)(46,54)(48,60)(49,53)(50,67)(51,59)(52,61)(55,57)(58,64)(62,70)(65,72)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1932);