#501622

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
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	#33582	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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)(5,6,11,15,12,16,9,10)(7,13,27,31,19,29,18,25)(8,20,23,14,24,26,22,30)(17,32,35,28,34,21,33,36);(1,2,4,3)(5,7)(6,13)(8,21)(9,18)(10,25)(11,27)(12,19)(14,17)(15,31)(16,29)(20,33)(22,28)(23,36)(24,32)(26,35)(30,34);(5,8,12,24)(6,14,16,30)(7,17,19,34)(9,23,11,22)(10,26,15,20)(13,28,29,36)(18,35,27,33)(21,31,32,25);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=i,d^2=e,f^2=i,g^2=hi,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, 408818);