#244147

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	#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

701 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efh,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, 151343);