#146305

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ci,d^b=di,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53501);