#489998

order	512
nilpotency class	4
isoclinism	isoclinic class of #489745
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	#61489	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

737 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)(9,10,15,21)(11,22,23,38)(12,26,30,37)(13,20,14,19)(16,32,34,24)(17,31,36,25)(18,28,33,29)(27,39,40,35);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,27)(13,25)(14,31)(15,23)(17,33)(18,36)(19,35)(20,39)(21,34)(22,26)(24,29)(28,32)(30,40)(37,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,24)(13,31)(14,25)(15,23)(16,33)(18,34)(19,38)(20,22)(21,36)(26,39)(27,29)(28,40)(30,32)(35,37);(1,3,5,4)(2,6,7,8)(11,23)(16,34)(17,36)(22,38)(24,32)(25,31)(27,40)(35,39)

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,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397194);