#7038725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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	#39004	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=fi,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=eh,d^c=dh,g^c=gi

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, 6945921);