#151092

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle, prime radical	C₂²	D₄×C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,21,26,22,27,35,42,23,19,28,30,36,31,43,45,20,24,32,37,33,38,46,47,34,29,25,39,40,44,41,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,24)(19,29)(26,37)(27,38)(30,39)(31,44)(42,47)(45,48);(17,19,21,30,22,31,35,45,23,24,28,37,36,38,43,47,20,29,32,39,33,44,46,48,34,18,25,26,40,27,41,42)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ef,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,c^b=ci,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58288);