#141413

order	512
nilpotency class	4
isoclinism	isoclinic class of #141367
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

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

normal subgroups only

This group has 277 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48609);