#6609026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#39052	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,23,26)(19,24,28,31)(20,27,30,32)(21,25,22,29);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,28)(18,24,26,31)(20,29,30,25)(21,32,22,27);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,22,31,23,30,21,24)(18,19,25,27,26,28,29,32);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,23,22)(18,25,26,29)(19,27,28,32)(20,24,30,31);(11,12)(14,16)(19,28)(20,30)(24,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6516222);