#506391

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28240	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 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:

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

Polycyclic presentation:

2^9|b^2=egh,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^c=fi

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, 413587);