#135071

order	512
nilpotency class	4
isoclinism	isoclinic class of #135037
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

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

normal subgroups only

This group has 240 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=i,d^2=gi,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=eh,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, 42267);