#509613

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 416809);