#481665

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×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

876 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,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, 388861);