#6826713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
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	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18)(19,27)(20,25)(21,23)(22,26)(24,30)(28,31)(29,32);(9,10)(11,14)(12,16)(13,15)(18,23)(19,28)(20,30)(22,29)(24,31)(25,27);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,29,20,21,30,22,28)(18,24,32,31,23,27,26,25);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(18,23)(24,27)(25,31)(26,32);(9,12,13,11)(10,14,15,16)(17,20,22,19,21,28,29,30)(18,25,26,27,23,31,32,24)

Polycyclic presentation:

2^9|a^2=fi,c^2=fh,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi

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, 6733909);