#206803

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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)(4,5)(6,8)(9,10)(11,18)(12,17)(13,22)(14,16)(15,19)(20,24)(21,23)(25,26,28,32,29,31,35,38)(27,30,33,37,34,36,39,40);(2,5)(6,7)(9,11,12,21,14,20,13,19)(10,15,22,24,16,23,17,18)(25,27,28,33,29,34,35,39)(26,30,32,37,31,36,38,40);(1,3,4,8)(2,6,5,7)(10,16)(15,23)(17,22)(18,24);(9,12,14,13)(10,17,16,22)(11,19,20,21)(15,24,23,18)(26,31)(30,36)(32,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh

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, 113999);