#239241

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146437);