#7148508

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

1696 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055704);