#505377

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

492 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)(4,5)(6,8)(10,16)(12,26)(14,27)(20,30)(21,38)(22,39)(24,40)(34,36);(2,5)(6,7)(9,10,15,30,13,16,28,20)(11,21,25,36,23,38,31,34)(12,18,14,29,26,19,27,35)(17,22,37,24,32,39,33,40);(9,11,13,23)(10,17,16,32)(12,24,26,40)(14,22,27,39)(15,31,28,25)(18,34,19,36)(20,37,30,33)(21,29,38,35);(1,3,4,8)(2,6,5,7)(9,12,15,14,13,26,28,27)(10,18,30,29,16,19,20,35)(11,22,25,24,23,39,31,40)(17,21,37,36,32,38,33,34)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=efhi,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, 412573);