#391654

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 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)(3,6)(4,5)(7,8)(9,10,16,20,14,21,15,31)(11,17,25,34,23,35,24,38)(12,26,28,18,13,19,30,36)(22,33,27,32,39,40,29,37);(3,7)(6,8)(9,11)(10,17)(12,27)(13,29)(14,23)(15,25)(16,24)(18,33)(19,37)(20,38)(21,35)(22,28)(26,32)(30,39)(31,34)(36,40);(2,5)(6,8)(9,12)(10,18)(11,22)(13,14)(15,30)(16,28)(17,32)(19,20)(21,36)(23,39)(24,29)(25,27)(26,31)(33,38)(34,40)(35,37);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,19,21,26)(11,22,23,39)(15,28,16,30)(17,33,35,40)(18,20,36,31)(24,29,25,27)(32,38,37,34)

Polycyclic presentation:

2^9|a^2=h,d^2=fg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=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, 298850);