#233623

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1328 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,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=dhi,e^b=eh,f^b=fh,d^c=dhi,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, 140819);