#7003058

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,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,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, 6910254);