#203710

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110906);