#220616

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61088	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127812);