#481689

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388885);