#7143217

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

1943 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)(11,23)(12,28)(13,19)(14,22)(15,21)(16,20)(17,26)(18,29)(24,33)(25,38)(27,37)(30,40)(31,35)(32,34)(36,39);(2,5)(3,7)(9,11)(10,17)(12,24)(13,30)(14,32)(15,26)(16,27)(18,33)(19,36)(20,34)(21,37)(22,23)(25,31)(28,38)(29,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,20,35)(11,24,27,30)(14,29,15,31)(17,33,34,40)(19,21,28,22)(23,36,37,38)(25,32,39,26);(2,5)(6,8)(9,13,16,12)(10,19,20,28)(11,25,27,39)(14,29,15,31)(17,33,34,40)(18,21,35,22)(23,38,37,36)(24,32,30,26);(1,3,4,7)(2,6,5,8)(10,20)(17,34)(18,35)(19,28)(21,22)(23,37)(33,40)(36,38)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,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, 7050413);