#148914

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

478 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)(6,13)(7,18)(8,9)(10,11)(14,30)(15,16)(17,29)(20,21)(22,34)(23,24)(25,26)(27,31)(28,32)(35,36);(5,6,9,25,11,16,24,29,12,17,23,15,10,26,8,13)(7,19,18,28,21,14,31,36,22,33,34,35,27,30,20,32);(5,7,8,20,10,27,23,34,12,22,24,31,11,21,9,18)(6,14,13,28,26,19,15,32,17,30,29,35,16,33,25,36)

Polycyclic presentation:

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

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, 56110);