#7185440

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	#60820	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(9,10,15,20)(11,21,25,33)(12,26,27,40)(13,18,30,39)(14,31,32,19)(16,24,34,23)(17,35,36,22)(28,38,29,37);(1,2,5,6)(3,4,8,7)(9,11,15,25)(10,16,20,34)(12,13,27,30)(14,24,32,23)(17,18,36,39)(19,33,31,21)(22,29,35,28)(26,38,40,37);(1,3)(2,7)(4,6)(5,8)(9,12,14,22,15,27,32,35)(10,17,19,26,20,36,31,40)(11,13,24,29,25,30,23,28)(16,18,33,38,34,39,21,37);(2,6)(3,8)(9,13)(10,18)(11,22)(12,23)(14,29)(15,30)(16,26)(17,21)(19,38)(20,39)(24,27)(25,35)(28,32)(31,37)(33,36)(34,40);(1,4,5,7)(2,3,6,8)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,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, 7092636);