#460616

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=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, 367812);