#481105

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3444	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

819 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=gi,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,e^b=egi,f^b=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388301);