#131697

order	512
nilpotency class	4
isoclinism	isoclinic class of #131664
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

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

normal subgroups only

This group has 506 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)(17,18,22,29,24,30,44,27,23,28,46,57,43,56,21,42)(19,31,34,49,36,55,67,60,35,58,69,25,47,50,33,51)(20,37,39,59,41,72,38,52,40,26,45,53,73,54,74,71)(32,62,64,70,66,79,63,80,65,61,68,48,76,77,75,78);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,16)(10,11)(17,19,23,35)(18,25,28,49)(20,32,40,65)(21,34,44,69)(22,33,46,67)(24,47,43,36)(26,48,37,70)(27,55,42,50)(29,58,57,31)(30,60,56,51)(38,68,74,64)(39,75,45,63)(41,76,73,66)(52,77,71,79)(53,61,59,62)(54,80,72,78);(1,4,7,2)(3,12,15,8)(5,10,6,9)(11,13,16,14)(17,20,23,40)(18,26,28,37)(19,32,35,65)(21,38,44,74)(22,45,46,39)(24,41,43,73)(25,48,49,70)(27,52,42,71)(29,59,57,53)(30,54,56,72)(31,61,58,62)(33,63,67,75)(34,68,69,64)(36,66,47,76)(50,79,55,77)(51,78,60,80)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,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, 38893);