#151743

order	512
nilpotency class	5
isoclinism	isoclinic class of #151734
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

162 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,29)(20,32)(21,38)(22,30)(23,28)(24,42)(25,31)(26,40)(27,35)(33,44)(34,41)(36,47)(37,39)(43,46)(45,48);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,21,31,23,33,40,24)(18,25,26,29,28,42,38,44)(20,34,27,37,36,30,43,45)(22,32,39,46,41,47,48,35);(3,10)(6,15)(9,16)(11,14)(17,20,35,38,24,37,48,25,40,43,47,28,33,34,41,44,23,36,46,26,31,45,39,42,21,27,32,18,19,30,22,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=d,c^2=eg,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,h^a=hi,c^b=cf

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58939);