#504262

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fh,d^2=e,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=gh,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, 411458);