#147390

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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	#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

700 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)(5,6,10,16,11,15,29,34,12,17,30,19,9,28,24,23)(7,18,22,27,21,32,36,25,20,13,31,26,14,33,35,8);(2,4)(5,7)(6,13,17,18)(8,23,25,34)(9,14)(10,22)(11,21)(12,20)(15,33,28,32)(16,26,19,27)(24,35)(29,36)(30,31);(5,8,24,33,9,26,30,13,12,25,29,32,11,27,10,18)(6,14,34,22,28,20,16,35,17,21,23,31,15,7,19,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 54586);