#7148098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10,15,19,14,21,13,20)(11,22,25,16,27,37,26,33)(12,18,28,17,30,36,29,34)(23,31,24,38,40,32,39,35);(2,5)(3,7)(9,11,14,27)(10,16,21,33)(12,23,30,40)(13,26,15,25)(17,31,34,32)(18,35,36,38)(19,37,20,22)(24,29,39,28);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,23)(13,28)(14,30)(15,29)(16,31)(18,19)(20,36)(21,34)(22,38)(24,26)(25,39)(27,40)(32,33)(35,37);(2,5)(6,8)(9,12)(10,18)(11,24)(13,29)(14,30)(15,28)(16,32)(17,19)(20,34)(21,36)(22,38)(23,26)(25,40)(27,39)(31,33)(35,37);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055294);