#137462

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂³	#442
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44658);