#491736

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	#19968	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^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.

Relationships

maximal subgroup of	#72309961, #72310003
maximal normal subgroup of	#72309961, #72310003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398932);