#239169

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
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

936 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)(7,10)(8,11)(9,12)(13,17)(14,19)(15,20)(16,18)(22,27)(23,30)(24,25)(28,35)(29,34)(32,33);(5,7,9,16)(6,10,12,18)(8,14,17,15)(11,19,13,20)(21,22,26,29)(23,31,33,36)(24,27,25,34)(28,32,35,30);(2,4)(6,11)(7,14)(8,17)(12,13)(15,16)(19,20)(21,23,25,30,26,33,24,32)(22,28,27,31,29,35,34,36);(6,12)(10,18)(11,13)(19,20)(21,24,26,25)(22,27,29,34)(23,32,33,30)(28,31,35,36)

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=di,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146365);