#509617

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

647 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 416813);