#243855

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
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

672 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151051);