#220614

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

1013 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)(11,20)(12,25,28,35)(13,29)(14,18)(15,19)(16,23)(17,27,36,31)(21,34,26,32)(22,24)(30,33)(37,40,38,39);(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,22)(19,33)(20,29)(21,28)(25,38)(27,39)(31,40)(32,36)(35,37);(2,5)(3,7)(9,12)(10,17)(11,21)(13,27)(14,31)(15,28)(16,32)(18,35)(19,36)(20,37)(22,38)(23,39)(24,26)(25,29)(30,40)(33,34);(9,10)(11,22)(12,17)(13,29)(14,18)(15,19)(16,30)(20,24)(21,38)(23,33)(25,27)(26,37)(28,36)(31,35)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=g,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, 127810);