#504469

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	#13313	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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)(3,6)(4,5)(7,8)(9,10,13,20,15,22,26,35,14,21,33,36,16,23,31,17)(11,24,27,37,32,19,12,29,28,18,34,38,25,39,30,40);(10,17)(11,25)(13,31)(15,16)(18,37)(19,29)(20,23)(21,35)(22,36)(24,38)(26,33)(27,34)(28,32)(39,40);(2,5)(6,8)(9,11)(10,18)(12,26)(13,27)(14,28)(15,32)(16,25)(17,29)(19,23)(20,38)(21,24)(22,39)(30,31)(33,34)(35,40)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,26)(13,32)(14,30)(15,27)(16,34)(17,29)(18,23)(20,37)(21,39)(22,24)(25,33)(28,31)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=e,d^2=fi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,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, 411665);