#7143221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050417);