#264005

order	512
nilpotency class	3
isoclinism	isoclinic class of #264003
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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171201);