#243846

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

831 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,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, 151042);