#243902

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	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

718 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

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, 151098);