#141827

order	512
nilpotency class	4
isoclinism	isoclinic class of #141707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#490
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 273 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:

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

Polycyclic presentation:

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

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, 49023);