#146949

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ci,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54145);