#251430

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45986	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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)(3,7)(4,6)(5,8)(9,10,13,17)(11,18,20,23)(12,22,14,16)(15,19,24,21)(25,26,31,35)(27,32,28,37)(29,34,30,39)(33,36,40,38);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,16)(13,17)(14,22)(15,19)(20,23)(21,24)(25,27,31,28)(26,32,35,37)(29,36,30,38)(33,34,40,39);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,24)(18,22)(25,28,31,27)(26,33,35,40)(29,38,30,36)(32,39,37,34);(2,6)(5,7)(9,12,13,14)(10,16,17,22)(11,19,20,21)(15,23,24,18)(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,40,37,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158626);