#140066

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=d,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=eg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47262);