#147883

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=hi,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=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,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, 55079);