#239068

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
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

994 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)(5,6,9,13)(7,14,15,11)(8,10,17,12)(16,19,20,18)(21,22,27,32)(23,29,34,36)(24,35,33,28)(25,30,26,31);(6,10)(7,15)(8,17)(11,19)(12,13)(14,18)(21,23)(22,28)(24,25)(26,33)(27,34)(29,30)(31,36)(32,35);(2,4)(5,7,8,16,9,15,17,20)(6,11,12,19,13,14,10,18)(21,24,25,23,27,33,26,34)(22,29,31,35,32,36,30,28);(7,15)(11,14)(16,20)(18,19)(21,25,27,26)(22,30,32,31)(23,24,34,33)(28,29,35,36)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146264);