#7182009

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

1738 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)(9,10,13,20,14,18,30,19)(11,21,25,33,26,37,24,15)(12,17,28,34,29,36,27,16)(22,31,39,35,40,32,23,38);(1,2,5,6)(3,4,8,7)(9,11)(10,15)(12,22)(13,25)(14,26)(16,31)(17,35)(18,33)(19,37)(20,21)(23,27)(24,30)(28,39)(29,40)(32,34)(36,38);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,22)(13,27)(14,29)(15,31)(17,19)(18,34)(20,36)(21,38)(23,25)(24,39)(26,40)(28,30)(32,33)(35,37);(2,6)(3,8)(9,12)(10,17)(11,23)(13,28)(14,29)(15,32)(16,19)(18,36)(20,34)(21,38)(22,25)(24,40)(26,39)(27,30)(31,33)(35,37);(1,4,5,7)(2,3,6,8)(10,18)(15,33)(16,34)(17,36)(19,20)(21,37)(31,32)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,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, 7089205);