#508425

order	512
nilpotency class	4
isoclinism	isoclinic class of #508289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 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:

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

Polycyclic presentation:

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

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, 415621);