#500468

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28240	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

771 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)(9,10)(11,21)(12,19)(13,18)(14,17)(15,30)(16,24)(20,27)(22,36)(23,32)(25,31)(26,34)(28,33)(29,35)(37,38)(39,40);(2,5)(6,7)(9,11,13,24)(10,16,18,21)(12,23,14,28)(15,22,27,39)(17,32,19,33)(20,36,30,40)(25,26,38,29)(31,35,37,34);(10,17)(11,22)(12,14)(15,29)(16,31)(18,19)(20,30)(21,37)(23,25)(24,39)(26,27)(28,38)(32,36)(33,40);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,17,18,19)(11,23,24,28)(15,26,27,29)(16,32,21,33)(20,34,30,35)(22,38,39,25)(31,40,37,36)

Polycyclic presentation:

2^9|b^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=di,f^c=fi,g^c=gh,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, 407664);