#504479

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

583 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=fhi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,e^c=ei,e^d=ehi,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411675);