#7148514

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

1710 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)(9,10,15,20,16,21,14,31)(11,22,26,17,25,35,32,34)(12,27,29,37,13,19,30,18)(23,36,24,39,28,33,40,38);(2,5)(3,7)(9,11,16,25)(10,17,21,34)(12,28,13,23)(14,32,15,26)(18,36,37,33)(19,38,27,39)(20,35,31,22)(24,29,40,30);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,23)(13,16)(14,29)(15,30)(17,33)(19,20)(21,37)(22,38)(24,32)(25,28)(26,40)(27,31)(34,36)(35,39);(2,5)(6,8)(9,13,16,12)(10,19,21,27)(11,24,25,40)(14,29,15,30)(17,33,34,36)(18,31,37,20)(22,39,35,38)(23,26,28,32);(1,3,4,7)(2,6,5,8)(11,25)(17,34)(22,35)(23,28)(24,40)(26,32)(33,36)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=hi,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, 7055710);