#287773

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 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)(5,6,9,13,8,12,15,11)(7,14,18,20,17,10,16,19)(21,22,25,28)(23,26,30,34)(24,27,31,35)(29,33,32,36);(1,2,4,3)(5,7,15,16,8,17,9,18)(6,10,11,20,12,14,13,19)(23,29)(24,31)(26,33)(27,35)(30,32)(34,36);(6,11)(7,16)(9,15)(10,14)(12,13)(17,18)(21,23,25,30)(22,26,28,34)(24,32,31,29)(27,36,35,33);(1,3,4,2)(6,12)(10,14)(11,13)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194969);