#502083

order	512
nilpotency class	4
isoclinism	isoclinic class of #502081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409279);