#149468

order	512
nilpotency class	4
isoclinism	isoclinic class of #129565
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

694 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56664);