#502196

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33526	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409392);