#501990

order	512
nilpotency class	4
isoclinism	isoclinic class of #501810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28233	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409186);