#134621

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41817);