#145522

order	512
nilpotency class	4
isoclinism	isoclinic class of #145502
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,25,32)(19,33,39,66)(20,40,47,63)(21,29,48,57)(22,28,42,62)(23,31,24,30)(26,50,55,35)(27,49,61,44)(34,67,72,51)(36,54,68,53)(37,65,38,52)(41,64,73,80)(43,60,76,59)(45,74,46,58)(56,75,79,69)(70,77,71,78);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,19,25,39)(18,26,32,55)(20,41,47,73)(21,36,48,68)(22,35,42,50)(23,38,24,37)(27,56,61,79)(28,52,62,65)(29,33,57,66)(30,54,31,53)(34,49,72,44)(40,51,63,67)(43,71,76,70)(45,75,46,69)(58,78,74,77)(59,80,60,64);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,20,42,44,23,45,21,43,25,47,22,49,24,46,48,76)(18,27,57,40,30,59,28,58,32,61,29,63,31,60,62,74)(19,34,68,41,37,70,35,69,39,72,36,73,38,71,50,75)(26,51,65,56,53,77,33,64,55,67,52,79,54,78,66,80)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,h^b=hi,e^c=eghi,f^c=fghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52718);