#509851

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
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 ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

819 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:

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

Polycyclic presentation:

2^9|a^2=efi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417047);