#6594043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18,20,22)(19,23,26,29)(21,24,27,30)(25,32,28,31);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,15)(14,16)(18,22)(23,29)(24,30)(31,32);(1,4,5,3)(2,6,8,7)(11,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6501239);