#131078

order	512
nilpotency class	4
isoclinism	isoclinic class of #130736
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3155	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 570 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) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38274);