#483177

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

767 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=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, 390373);