#491740

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19971	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398936);