#332408

order	512
nilpotency class	3
isoclinism	isoclinic class of #332327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

645 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239604);