#231365

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=dhi

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, 138561);