#146121

order	512
nilpotency class	4
isoclinism	isoclinic class of #146120
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

189 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=dhi,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,d^c=di,e^c=eh,f^c=fgi,h^c=hi

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, 53317);