#139419

order	512
nilpotency class	4
isoclinism	isoclinic class of #139358
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#416
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=h,d^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46615);