#7117019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#23972	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=gh,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^c=ei,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, 7024215);