#332680

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
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	#24227	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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)(11,21)(12,18)(13,16)(14,19)(15,20)(17,22)(23,37)(24,29)(25,34)(26,30)(27,33)(28,32)(31,38)(35,39)(36,40);(10,16)(11,22)(12,25)(13,14)(18,33)(19,20)(21,29)(24,28)(26,27)(31,37)(35,40)(36,38);(2,5)(3,8)(6,7)(9,11)(10,17)(12,23)(13,22)(14,28)(15,24)(16,29)(18,31)(19,21)(20,32)(25,38)(26,36)(27,39)(30,37)(33,40)(34,35);(2,5)(3,8)(6,7)(9,12,15,27)(10,18,20,33)(11,23,24,39)(13,26,14,25)(16,30,19,34)(17,31,32,40)(21,35,29,37)(22,36,28,38)

Polycyclic presentation:

2^9|a^2=hi,d^2=h,e^2=h,f^2=i,g^2=h,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,g^b=gh,f^c=fi,g^c=gh,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, 239876);