#244153

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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	#2397
prime radical	#10012	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151349);