#223143

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

772 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)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24)(25,26)(27,33,34,40)(28,30)(29,32)(31,35,38,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24)(26,30)(31,38)(32,39)(33,40);(1,3,4,7)(2,6,5,8)(10,15)(12,19)(14,20)(17,23)(25,27,28,34)(26,31,30,38)(29,37,36,35)(32,40,39,33);(2,5)(6,8)(26,30)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,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, 130339);