#127936

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35132);