#6628663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

1497 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(17,18)(19,26)(20,25)(21,23)(22,30)(24,28)(27,32)(29,31);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(18,23)(19,27)(25,30)(28,29);(9,11,13,12)(10,15,14,16)(17,19,22,28,21,27,20,29)(18,24,25,26,23,31,30,32);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,18)(19,24)(20,30)(21,23)(22,25)(26,28)(27,31)(29,32);(9,12,13,11)(10,16,14,15)(17,20,21,22)(18,25,23,30)(19,28,27,29)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=ghi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6535859);