#217089

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

739 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(25,26,30,32,28,33,29,34)(27,35,38,39,36,31,37,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,23)(16,21)(17,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,35)(34,39);(2,5)(6,8)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(26,32)(27,36)(29,30)(31,39)(33,34)(35,40);(1,3,4,7)(2,6,5,8)(10,14)(15,23)(16,21)(17,24)(26,33)(27,36)(32,34)(37,38)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ei,f^2=i,g^2=hi,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, 124285);