#137643

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

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	#3035	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

377 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)(3,6)(7,8)(9,10,15,19,14,18,22,23,16,20,35,39,13,32,33,28)(11,21,17,36,26,31,38,34,25,29,37,40,24,12,27,30);(1,2,6,8,4,7,3,5)(9,11,22,38,13,24,15,17,16,25,33,27,14,26,35,37)(10,12,28,40,32,29,39,34,20,31,23,36,18,21,19,30);(2,7)(5,8)(9,12)(10,17)(11,23)(13,29)(14,21)(15,34)(16,31)(18,38)(19,24)(20,37)(22,40)(25,28)(26,39)(27,32)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 44839);