#7143614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2135 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)(10,15)(11,19)(13,27)(17,29)(18,30)(20,35)(21,22)(23,34)(24,25)(28,32)(36,39)(37,38);(2,5)(3,7)(9,10,13,15,14,18,27,30)(11,16,21,28,22,33,19,32)(12,23,24,34,26,29,25,17)(20,36,35,40,37,39,38,31);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,16,18,33)(12,20,26,37)(13,21,27,19)(15,28,30,32)(17,31,34,40)(23,36,29,39)(24,35,25,38);(2,5)(6,8)(9,12)(10,17)(11,20)(13,25)(14,26)(15,29)(16,31)(18,34)(19,35)(21,38)(22,37)(23,30)(24,27)(28,39)(32,36)(33,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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,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, 7050810);