#147131

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 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)(6,12,14,20)(13,27,29,16)(15,26,31,35)(30,36,32,33);(5,6,9,15,10,12,24,26,11,14,25,31,8,20,22,35)(7,16,18,33,17,29,34,32,19,27,23,36,21,13,28,30);(2,4)(5,7)(6,13,14,29)(8,21)(9,23)(10,17)(11,19)(12,16,20,27)(15,32,31,30)(18,25)(22,34)(24,28)(26,36,35,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di,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, 54327);