#216863

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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,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, 124059);