#489718

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
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	#61086	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

893 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)(3,6)(4,7)(5,8)(9,10,14,19)(11,20,23,39)(12,17,15,24)(13,26,28,37)(16,31,33,22)(18,25,36,29)(21,38,30,35)(27,32,40,34);(2,6)(3,4)(7,8)(9,11,14,23)(10,16,19,33)(12,21,15,30)(13,27,28,40)(17,32,24,34)(18,35,36,38)(20,37,39,26)(22,25,31,29);(2,7)(3,4)(6,8)(9,10)(11,16)(12,24)(13,18)(14,19)(15,17)(20,22)(21,34)(23,33)(25,37)(26,29)(27,35)(28,36)(30,32)(31,39)(38,40);(2,8)(6,7)(9,12,14,15)(10,17,19,24)(11,21,23,30)(13,25,28,29)(16,32,33,34)(18,26,36,37)(20,38,39,35)(22,27,31,40)

Polycyclic presentation:

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

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, 396914);