#7081590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988786);