#147053

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#476
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54249);