#489960

order	512
nilpotency class	4
isoclinism	isoclinic class of #489701
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	#61436	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

754 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397156);