#131074

order	512
nilpotency class	4
isoclinism	isoclinic class of #130730
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi

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, 38270);