#481131

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

961 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=gi,f^2=h,h^2=i,b^a=be,c^a=cgi,d^a=di,e^a=eg,d^b=di,e^b=egi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388327);