#7078534

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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	#25549	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(18,23)(20,21)(24,29)(25,31)(26,32)(27,30);(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,23)(21,29)(22,24)(25,27)(28,31)(30,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(18,24)(23,29)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,19,22,28)(18,25,24,32)(20,27,21,30)(23,31,29,26);(10,14)(13,15)(17,20,22,21)(18,23,24,29)(19,27,28,30)(25,31,32,26)

Polycyclic presentation:

2^9|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,f^b=fi,d^c=dhi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6985730);