#491115

order	512
nilpotency class	4
isoclinism	isoclinic class of #491113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19329	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

453 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)(3,5)(4,6)(7,8)(10,15)(12,23)(13,24)(16,25)(17,26)(18,27)(19,30)(21,36)(22,35)(28,29)(31,34)(32,40)(33,39)(37,38);(9,10,16,30,13,19,12,15,14,18,23,29,24,28,25,27)(11,17,31,40,22,32,21,26,20,33,36,38,35,37,34,39);(2,5)(3,8)(6,7)(9,11)(10,17)(12,21)(13,22)(14,20)(15,26)(16,31)(18,33)(19,32)(23,36)(24,35)(25,34)(27,39)(28,37)(29,38)(30,40);(2,6)(5,7)(10,18)(11,20)(15,27)(19,28)(21,34)(22,35)(29,30)(31,36)

Polycyclic presentation:

2^9|b^2=egi,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,d^c=di,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398311);