#7003079

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

1832 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)(9,10,13,20,14,18,29,19)(11,21,24,15,25,32,38,31)(12,17,27,33,28,35,26,16)(22,37,23,36,40,30,39,34);(1,2)(3,7)(4,6)(5,8)(9,11,13,24,14,25,29,38)(10,15,19,21,18,31,20,32)(12,22,26,39,28,40,27,23)(16,30,17,34,33,37,35,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,22)(13,26)(14,28)(15,30)(17,19)(18,33)(20,35)(21,34)(23,38)(24,39)(25,40)(27,29)(31,37)(32,36);(3,8)(5,7)(9,12)(10,17)(11,23)(13,27)(14,28)(15,30)(16,19)(18,35)(20,33)(21,36)(22,38)(24,40)(25,39)(26,29)(31,37)(32,34);(2,6)(5,7)(10,18)(15,31)(16,33)(17,35)(19,20)(21,32)(30,37)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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,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, 6910275);