#148268

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

836 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55464);