#217063

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=ehi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124259);