#7047435

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6954631);