#137635

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,e^b=ei,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44831);