#503437

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61038	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

985 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)(4,6)(5,8)(9,10)(11,22)(12,26)(13,19)(14,32)(15,34)(16,35)(17,24)(18,30)(20,27)(21,23)(25,38)(28,40)(29,39)(31,37)(33,36);(1,3,5,4)(2,6,8,7)(9,11,13,24)(10,17,19,22)(12,27,29,28)(14,33,15,31)(16,30,23,25)(18,35,38,21)(20,26,40,39)(32,37,34,36);(1,3,5,4)(2,7,8,6)(9,12,28,24,16,31,14,30,13,29,27,11,23,33,15,25)(10,18,34,37,21,17,20,26,19,38,32,36,35,22,40,39);(1,4,5,3)(2,7,8,6)(10,19)(17,22)(18,38)(20,40)(21,35)(26,39)(32,34)(36,37)

Polycyclic presentation:

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

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, 410633);