#148755

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,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, 55951);