#7185444

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
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	#24231	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7092640);