#332347

order	512
nilpotency class	3
isoclinism	isoclinic class of #332327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18766	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

712 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)(3,5)(4,7)(6,8)(9,10)(11,20)(12,16)(13,17)(14,19)(15,23)(18,28)(21,36)(22,30)(24,34)(25,35)(26,31)(27,33)(29,37)(32,39)(38,40);(9,11,14,24)(10,15,19,30)(12,25,27,21)(13,22,28,23)(16,29,33,40)(17,20,18,34)(26,38,39,37)(31,35,32,36);(2,5)(3,8)(6,7)(9,12,14,27)(10,16,19,33)(11,21,24,25)(13,26,28,39)(15,29,30,40)(17,31,18,32)(20,35,34,36)(22,37,23,38);(2,5)(3,8)(6,7)(10,17)(11,22)(13,28)(16,31)(18,19)(20,34)(21,37)(23,24)(25,38)(26,39)(32,33)(35,36)

Polycyclic presentation:

2^9|b^2=h,c^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, 239543);