#491107

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
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

450 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)(12,24)(13,25)(16,26)(17,27)(18,31)(19,30)(20,35)(22,33)(23,38)(28,29)(32,37)(34,40)(36,39);(9,10,16,31,13,18,12,15,14,19,24,29,25,28,26,30)(11,20,32,39,23,36,22,35,21,17,33,40,38,34,37,27);(2,5)(3,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,21)(15,27)(16,32)(18,34)(19,20)(24,33)(25,38)(26,37)(28,36)(29,39)(30,35)(31,40);(2,6)(5,7)(11,21)(17,20)(22,37)(23,38)(27,35)(32,33)(34,36)(39,40)

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,c^b=ci,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398303);