#491772

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
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

458 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)(9,10,13,18)(11,20,15,16)(12,23,24,39)(14,27,26,32)(17,33,34,25)(19,22,35,29)(21,38,30,36)(28,37,40,31);(1,2)(3,6)(4,7)(5,8)(9,11,13,15)(10,16,18,20)(12,21,24,30)(14,22,26,29)(17,31,34,37)(19,32,35,27)(23,36,39,38)(25,28,33,40);(9,12)(10,17)(11,21)(13,24)(14,28)(15,30)(16,31)(18,34)(19,36)(20,37)(22,33)(23,27)(25,29)(26,40)(32,39)(35,38);(2,6)(3,8)(5,7)(12,24)(17,34)(21,30)(23,39)(25,33)(28,40)(31,37)(36,38)

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=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309961
maximal normal subgroup of	#72309961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398968);