#6622972

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	#33183	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(4,6)(5,8)(10,14)(15,16)(17,18,21,24)(19,26,28,32)(20,23,30,25)(22,27,31,29);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,18)(19,22)(20,25)(21,24)(23,30)(26,29)(27,32)(28,31);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19,20,29,21,28,30,27)(18,22,25,26,24,31,23,32);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(17,20,21,30)(18,23,24,25)(19,27,28,29)(22,26,31,32);(9,12,13,11)(10,16,14,15)(19,28)(22,31)(26,32)(27,29)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=fh,e^2=h,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=eh

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, 6530168);