#148955

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
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

450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 56151);