#149076

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
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

464 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)(5,6,11,16,13,22,29,20,12,17,9,15,10,7,18,21)(8,23,25,24,26,35,32,34,28,19,30,36,27,14,31,33);(5,7,12,22)(6,13,17,10)(8,24)(9,16,11,15)(14,32)(18,21,29,20)(19,30)(23,25)(26,33)(27,34)(28,36)(31,35);(5,8,9,25,13,26,18,32,12,28,11,30,10,27,29,31)(6,14,15,33,22,23,21,24,17,35,16,34,7,19,20,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=hi,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=fhi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,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, 56272);