#483175

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

778 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)(9,10,14,20,29,36,27,37,15,17,13,19,12,18,28,38)(11,21,24,35,22,40,30,16,26,34,23,33,25,39,31,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,30)(14,31)(15,26)(17,32)(18,33)(19,39)(20,40)(21,37)(23,27)(24,28)(25,29)(34,38)(35,36);(1,3,5,4)(2,6,7,8)(10,17)(16,32)(18,36)(19,20)(21,34)(33,35)(37,38)(39,40);(9,12,15,29)(10,18,17,36)(11,22,26,25)(13,27,14,28)(16,33,32,35)(19,37,20,38)(21,40,34,39)(23,31,24,30)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cgi,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, 390371);