#491745

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

486 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)(11,20)(12,18)(13,16)(14,26)(15,29)(17,23)(19,28)(21,37)(22,36)(24,32)(25,39)(27,33)(30,35)(31,38)(34,40);(1,2)(3,6)(4,7)(5,8)(10,16)(17,31)(18,29)(19,32)(20,36)(26,39)(33,34)(35,37);(9,11)(10,17)(12,21)(13,22)(14,27)(15,30)(16,31)(18,33)(19,35)(20,26)(23,28)(24,38)(25,40)(29,34)(32,37)(36,39);(2,6)(3,8)(5,7)(9,12,13,15)(10,18,16,29)(11,21,22,30)(14,24,25,28)(17,33,31,34)(19,26,32,39)(20,37,36,35)(23,27,38,40)

Polycyclic presentation:

2^9|d^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,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	#72287259, #72310003
maximal normal subgroup of	#72287259, #72310003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398941);