#187472

order	512
nilpotency class	3
isoclinism	isoclinic class of #187410
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	#62833	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2220 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:

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

Polycyclic presentation:

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

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, 94668);