#187889

order	512
nilpotency class	3
isoclinism	isoclinic class of #187872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62822	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2278 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,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, 95085);