#140582

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

311 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,21,26,22,27,37,48,23,28,38,49,39,50,63,74)(19,24,31,42,32,43,57,69,30,41,55,67,56,68,78,80)(20,25,35,46,36,47,52,65,34,45,59,71,62,73,77,75)(29,51,54,76,33,58,61,79,53,40,64,66,60,44,70,72);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,21,31,22,32,37,57,23,30,38,55,39,56,63,78)(18,24,26,42,27,43,48,69,28,41,49,67,50,68,74,80)(20,33,59,70,36,53,77,54,34,60,35,61,62,29,52,64)(25,44,71,79,47,51,75,66,45,58,46,72,73,40,65,76);(2,7)(6,14)(8,12)(9,15)(17,20)(18,25)(19,29,30,53)(21,35)(22,36)(23,34)(24,40,41,51)(26,46)(27,47)(28,45)(31,54,55,64)(32,33,56,60)(37,52)(38,59)(39,62)(42,66,67,76)(43,44,68,58)(48,65)(49,71)(50,73)(57,61,78,70)(63,77)(69,72,80,79)(74,75)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ch,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47778);