#243873

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8,6,3,4,7)(9,10,14,19,15,20,13,18)(11,21,23,31,24,35,29,16)(12,25,26,34,27,17,28,33)(22,36,38,32,39,40,37,30);(1,2)(3,6)(4,8)(5,7)(9,11,13,23,15,24,14,29)(10,16,18,21,20,31,19,35)(12,22,28,38,27,39,26,37)(17,32,34,40,25,30,33,36);(1,3,6,2)(4,7,5,8)(9,12,15,27)(10,17,20,25)(11,22,24,39)(13,28,14,26)(16,30,31,32)(18,34,19,33)(21,36,35,40)(23,38,29,37);(1,4,6,5)(2,7,3,8)(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,26,27,28)(16,21,31,35)(17,33,25,34)(22,37,39,38)(30,40,32,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,d^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,d^c=dh,e^c=ei,f^c=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, 151069);