#509178

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

771 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=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, 416374);