#342519

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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)(3,7)(4,6)(5,8)(9,10,14,15,13,17,11,18)(12,20,19,24,21,23,22,16)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,39)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,23,24,20)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(9,12,13,21)(10,16,17,24)(11,19,14,22)(15,23,18,20)(25,27,29,34)(26,30,31,38)(28,36,35,33)(32,40,39,37);(2,6)(3,8)(10,15)(11,14)(16,23)(17,18)(19,22)(20,24)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 249715);