#481668

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

872 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)(9,10,14,20,15,19,11,16)(12,23,21,36,25,17,29,30)(13,18,27,35,28,34,22,31)(24,33,37,40,26,32,38,39);(1,2)(3,7)(4,6)(5,8)(9,11,15,14)(10,16)(12,21,25,29)(13,22,28,27)(17,30)(18,31)(19,20)(23,36)(24,37,26,38)(32,39)(33,40)(34,35);(9,12,15,25)(10,17,19,23)(11,21,14,29)(13,26,28,24)(16,30,20,36)(18,33,34,32)(22,38,27,37)(31,40,35,39);(9,13,15,28)(10,18,19,34)(11,22,14,27)(12,24,25,26)(16,31,20,35)(17,32,23,33)(21,37,29,38)(30,39,36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,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, 388864);