#7410067

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300884
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	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,3,6,2)(4,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(10,12)(11,15)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,4,6,5)(2,7,3,8)(9,12,13,10)(11,14,15,16)(19,27)(22,26)(28,32)(30,31);(1,5,6,4)(2,7,3,8)(11,15)(14,16)(18,24)(22,26)(23,25)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi

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, 7317263);