#6461260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368456);