#289474

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

708 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)(5,6,9,14,10,15,8,13)(7,11,16,12,18,20,17,19)(21,22,27,33,26,24,25,29)(23,28,34,36,30,32,31,35);(1,2,4,3)(5,7,8,17,10,18,9,16)(6,11,13,19,15,20,14,12)(21,23,25,31,26,30,27,34)(22,28,29,35,24,32,33,36);(5,7,10,18)(6,12,15,19)(8,16,9,17)(11,13,20,14)(21,24,26,22)(23,28,30,32)(25,29,27,33)(31,36,34,35);(1,3,4,2)(5,8,10,9)(6,13,15,14)(7,16,18,17)(11,12,20,19)(23,30)(28,32)(31,34)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=h,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, 196670);