#7181997

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=hi,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,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, 7089193);