#208487

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 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)(3,6)(4,5)(7,8)(10,14,16,19)(11,17)(15,22,23,21)(18,24)(25,26,30,32,29,34,28,33)(27,35,36,31,38,40,37,39);(9,10,12,19,13,16,20,14)(11,15,18,22,17,23,24,21)(25,27)(26,31)(28,36)(29,38)(30,37)(32,35)(33,40)(34,39);(2,5)(6,8)(9,11,13,17)(10,15,16,23)(12,18,20,24)(14,21,19,22)(25,28,29,30)(26,32,34,33)(27,36,38,37)(31,35,39,40);(1,3,4,7)(2,6,5,8)(11,17)(15,23)(18,24)(21,22)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115683);