#7118015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1423 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)(10,14)(11,12)(15,16)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(1,3,5,4)(2,6,8,7)(9,10,12,14,13,16,11,15)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,28)(29,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,16,15)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31);(10,15)(11,12)(14,16)(20,28)(23,32)(25,27)(29,31);(1,4,5,3)(2,7,8,6)(9,11,13,12)(10,14,16,15)(18,24)(19,26)(23,32)(25,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=ghi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei,f^d=fi

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, 7025211);