#504169

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×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 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)(9,10,14,20)(11,22,26,18)(12,27,28,39)(13,17,31,35)(15,21,16,32)(19,24,36,29)(23,38,30,37)(25,40,33,34);(1,3,5,4)(2,6,7,8)(10,17)(11,23)(13,29)(15,16)(18,27)(19,21)(20,35)(22,39)(24,31)(25,33)(26,30)(32,36)(34,37)(38,40);(1,4,5,3)(2,6,7,8)(9,11)(10,18)(12,24)(13,25)(14,26)(15,30)(16,23)(17,34)(19,27)(20,22)(21,38)(28,29)(31,33)(32,37)(35,40)(36,39);(9,12)(10,18)(11,24)(13,30)(14,28)(15,25)(16,33)(17,27)(19,34)(20,22)(21,37)(23,31)(26,29)(32,38)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,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, 411365);