#504163

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411359);