#234251

order	512
nilpotency class	3
isoclinism	isoclinic class of #234235
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

1348 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:

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

Polycyclic presentation:

2^9|b^2=h,c^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=dhi,e^b=eh,f^b=fi,e^c=eh,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, 141447);