#219438

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di

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, 126634);