#135151

order	512
nilpotency class	4
isoclinism	isoclinic class of #135117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#478
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,c^2=i,d^2=gi,e^2=ghi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42347);