#505657

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
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	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412853);