#234275

order	512
nilpotency class	3
isoclinism	isoclinic class of #234257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1290 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141471);