#148925

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56121);