#198200

order	512
nilpotency class	3
isoclinism	isoclinic class of #198126
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1263 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105396);