#440711

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 347907);