#6612640

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#32939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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)(12,15)(13,16)(17,18,23,26)(19,21,25,22)(20,28,31,24)(27,32,29,30);(9,11)(10,13)(12,16)(14,15)(17,19,22,18,23,25,21,26)(20,29,32,28,31,27,30,24);(2,5)(6,8)(11,16)(13,14)(17,20)(18,24)(19,27)(21,30)(22,32)(23,31)(25,29)(26,28);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,22)(18,25,26,19)(20,30,31,32)(24,29,28,27);(11,16)(13,14)(20,31)(24,28)(27,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519836);