#289450

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

692 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=hi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196646);