#501695

order	512
nilpotency class	4
isoclinism	isoclinic class of #501695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408891);