#505448

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

400 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)(3,7)(4,8)(5,6)(9,10)(11,21)(12,28)(13,17)(14,31)(15,20)(16,25)(18,29)(19,30)(22,34)(23,40)(24,36)(26,37)(27,39)(32,35)(33,38);(2,6)(7,8)(9,11,22,27,13,25,15,33)(10,16,34,38,17,21,20,39)(12,24,30,23,29,26,14,32)(18,36,31,40,28,37,19,35);(1,3,5,4)(2,7,6,8)(10,17)(11,23)(12,14)(15,22)(16,35)(18,31)(19,28)(21,40)(24,27)(25,32)(26,33)(29,30)(36,39)(37,38);(1,4,5,3)(2,7,6,8)(9,12,15,14,13,29,22,30)(10,18,20,19,17,28,34,31)(11,24,33,32,25,26,27,23)(16,36,39,35,21,37,38,40)

Polycyclic presentation:

2^9|b^2=fh,c^2=ei,d^2=efh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412644);