#147879

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1198 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)(9,10,13,18,14,20,31,19)(11,21,23,36,24,38,30,37)(12,25,28,17,29,16,27,35)(15,26,32,22,34,39,33,40);(2,4)(5,6)(7,8)(9,11,14,24)(10,15,20,34)(12,26,29,39)(13,30,31,23)(16,21,25,38)(17,36,35,37)(18,33,19,32)(22,28,40,27);(6,8)(10,16)(12,27)(13,31)(15,21)(17,18)(19,35)(20,25)(22,26)(23,30)(28,29)(32,37)(33,36)(34,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,g^b=gi,h^b=hi,d^c=dhi,e^c=ehi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55075);