#7148509

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

1691 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)(9,10,14,21,15,19,32,20)(11,22,26,34,27,37,25,16)(12,28,30,35,13,18,29,17)(23,33,24,38,31,40,39,36);(2,5)(3,7)(9,11)(10,16)(12,23)(13,31)(14,26)(15,27)(17,33)(18,36)(19,34)(20,37)(21,22)(24,30)(25,32)(28,38)(29,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,35)(11,23,27,31)(14,29,32,30)(16,33,34,40)(18,20,28,21)(22,36,37,38)(24,26,39,25);(2,5)(6,8)(9,13,15,12)(10,18,19,28)(11,24,27,39)(14,29,32,30)(16,33,34,40)(17,20,35,21)(22,38,37,36)(23,26,31,25);(1,3,4,7)(2,6,5,8)(10,19)(16,34)(17,35)(18,28)(20,21)(22,37)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=g,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,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, 7055705);