#410403

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317599);