#243863

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,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, 151059);