#483887

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#442
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391083);