#216311

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

861 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)(11,17)(12,20,22,15)(13,16)(14,19)(18,23,21,24)(25,26,30,34,29,32,28,33)(27,35,38,39,37,31,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,29,37)(26,31,32,35)(28,36,30,38)(33,39,34,40);(2,5)(6,8)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(26,32)(28,30)(31,35)(36,38);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,20)(23,24)(25,28,29,30)(26,33,32,34)(27,36,37,38)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=ehi,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123507);