#137604

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
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×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 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)(9,10,14,20,15,21,34,25,16,22,35,39,13,19,24,37)(11,23,18,38,26,12,28,30,27,29,32,40,31,17,36,33);(1,2,6,8,4,7,3,5)(9,11,24,28,13,31,35,18,16,27,34,36,15,26,14,32)(10,17,37,38,19,29,39,33,22,12,25,40,21,23,20,30);(9,12)(10,18)(11,25)(13,23)(14,33)(15,29)(16,17)(19,36)(20,31)(21,28)(22,32)(24,40)(26,39)(27,37)(30,35)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44800);