#6874072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
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	#36374	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6781268);