#7185429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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	#60817	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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)(11,23)(12,27)(13,19)(14,21)(15,20)(16,22)(17,25)(18,24)(26,39)(28,36)(29,37)(30,35)(31,40)(32,34)(33,38);(1,2,5,6)(3,4,8,7)(9,11)(10,17)(12,13)(14,28)(15,25)(16,26)(18,34)(19,37)(20,39)(21,23)(22,36)(24,31)(27,40)(29,33)(30,32)(35,38);(1,3)(2,7)(4,6)(5,8)(9,12,14,29,16,30,15,24)(10,18,20,35,22,37,21,27)(11,13,28,33,26,32,25,31)(17,34,39,38,36,19,23,40);(2,6)(3,8)(9,13)(10,19)(11,24)(12,28)(14,33)(15,31)(16,32)(17,35)(18,23)(20,40)(21,38)(22,34)(25,30)(26,29)(27,36)(37,39);(1,4,5,7)(2,3,6,8)

Polycyclic presentation:

2^9|b^2=h,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=dh,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, 7092625);