#215592

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

919 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)(11,20)(12,16)(13,18)(14,19)(15,23)(17,22)(21,24)(25,26,30,34,29,31,28,33)(27,35,37,32,36,40,38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,14)(13,22)(15,21)(17,23)(18,24)(19,20)(26,31)(27,36)(33,34)(37,38);(3,8)(5,7)(9,12,14,11)(10,17,19,24)(13,22,15,21)(16,18,20,23)(25,27)(26,32)(28,37)(29,36)(30,38)(31,39)(33,40)(34,35);(2,6)(5,7)(9,13,14,15)(10,18,19,23)(11,21,12,22)(16,17,20,24)(25,28,29,30)(26,33,31,34)(27,37,36,38)(32,40,39,35)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=h,d^2=i,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, 122788);