#200768

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107964);