#263293

order	512
nilpotency class	3
isoclinism	isoclinic class of #263291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170489);