#498146

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

794 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dehi,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405342);