#131701

order	512
nilpotency class	4
isoclinism	isoclinic class of #131668
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 507 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:

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

Polycyclic presentation:

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

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, 38897);