#287816

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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 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)(5,6,10,12,8,13,9,14)(7,15,16,19,17,11,18,20)(21,22,27,33,26,24,25,29)(23,28,34,36,31,32,30,35);(1,2,4,3)(5,7,10,16,8,17,9,18)(6,11,12,20,13,15,14,19)(21,23,27,34,26,31,25,30)(22,28,33,36,24,32,29,35);(6,12)(7,16)(9,10)(11,15)(13,14)(17,18)(21,24,26,22)(23,28,31,32)(25,29,27,33)(30,36,34,35);(1,3,4,2)(6,13)(11,15)(12,14)(19,20)

Polycyclic presentation:

2^9|a^2=fh,b^2=efi,c^2=i,d^2=ehi,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, 195012);