#502159

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409355);