#149074

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
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

476 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)(5,6,9,15,10,16,11,30,13,18,28,21,12,7,19,17)(8,20,26,36,29,14,31,22,27,35,23,33,24,34,25,32);(5,7)(6,12)(8,22,27,32)(9,21)(10,18)(11,30)(13,16)(14,23,34,26)(15,28)(17,19)(20,25,35,31)(24,33,29,36);(5,8,9,23,10,29,11,25,13,27,28,26,12,24,19,31)(6,14,15,32,16,35,30,36,18,34,21,22,7,20,17,33)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,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, 56270);