#7182006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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	#30866	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089202);