#339728

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246924);