#6611813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#23929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1932 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)(11,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,30)(22,26)(23,28)(27,32)(29,31);(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,30)(25,32)(26,31);(2,5)(6,8)(10,15)(11,12)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,31)(26,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(19,27)(23,31)(24,32)(28,29);(11,12)(14,16)(17,20,22,21)(18,25,26,30)(19,28,27,29)(23,32,31,24)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519009);