#243848

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

713 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:

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

Polycyclic presentation:

2^9|a^2=eh,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=dh,e^b=eh,f^b=fh,d^c=dh,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, 151044);