#151742

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

156 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,29)(20,24)(21,38)(22,41)(23,42)(25,32)(26,30)(27,39)(28,37)(31,49)(33,62)(34,63)(35,64)(36,66)(40,69)(43,57)(44,54)(45,65)(46,72)(47,73)(48,74)(50,53)(51,60)(52,59)(55,68)(56,67)(58,79)(61,70)(71,80)(75,77)(76,78);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,24)(25,43)(26,44)(27,45)(28,42)(29,49)(38,64)(41,66)(46,71)(47,70)(48,67)(50,75)(51,76)(52,63)(62,79)(68,69);(3,10)(6,15)(9,16)(11,14)(17,19,30,53,22,33,55,73,37,59,74,80,65,78,35,57,20,31,54,77,36,58,40,61,23,34,56,72,39,60,21,32)(18,25,38,51,27,46,67,63,42,70,69,79,66,75,44,49,24,43,64,76,45,71,48,52,28,47,68,62,41,50,26,29)

Polycyclic presentation:

2^9|a^2=f,b^2=di,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, 58938);