#147415

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

780 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)(3,4)(6,8)(9,10,14,19,16,12,28,21)(11,22,25,35,27,39,40,38)(13,30,31,18,32,20,15,29)(17,33,34,37,23,26,36,24);(2,4)(9,11,15,26,14,25,13,24,16,27,31,33,28,40,32,37)(10,17,20,35,21,36,18,22,12,23,30,38,19,34,29,39);(5,6,7,8)(9,12)(10,16)(11,23)(13,29)(14,19)(15,30)(17,27)(18,32)(20,31)(21,28)(22,37)(24,39)(25,34)(26,38)(33,35)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54611);