#146746

order	512
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

970 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)(5,6,11,18)(7,20)(8,26)(9,28,10,30)(12,19,13,32)(14,23)(15,25)(16,29,17,21)(22,34)(24,36)(27,33)(31,35);(2,4)(5,7,12,24,11,25,13,31)(6,14,19,33,18,34,32,26)(8,21,23,9,27,29,22,10)(15,17,35,30,20,16,36,28);(5,8)(6,15)(7,21)(9,24)(10,31)(11,27)(12,23)(13,22)(14,17)(16,34)(18,20)(19,35)(25,29)(26,28)(30,33)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53942);