#489992

order	512
nilpotency class	4
isoclinism	isoclinic class of #489733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61442	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

729 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fhi,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 397188);