#223155

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

783 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)(6,8)(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22)(25,26)(27,34)(28,36,37,39)(29,33)(30,31)(32,35,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,39,40,36);(1,3,4,7)(2,6,5,8)(10,15)(13,20)(14,17)(19,21)(25,28,29,37)(26,32,33,40)(27,35,30,38)(31,39,34,36);(2,5)(6,8)(11,18)(14,17)(19,21)(22,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,39,40,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=eh,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dei,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130351);