#243874

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:

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

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=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, 151070);