#259367

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10046	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

918 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)(4,5)(6,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,26,28,32)(27,34,36,30)(29,31,38,33)(35,40,37,39);(2,5)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,22)(18,24)(25,27)(26,30)(28,36)(29,35)(31,39)(32,34)(33,40)(37,38);(10,16)(12,13)(15,22)(17,18)(19,21)(23,24)(26,31)(27,35)(29,38)(32,33)(36,37)(39,40);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

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

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, 166563);