#446557

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=eg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=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, 353753);