#146066

order	512
nilpotency class	4
isoclinism	isoclinic class of #146024
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,22,27,42,56,73,69,23,28,45,59,21,26,41,55)(19,29,32,61,31,50,66,57,34,49,43,77,33,24,46,48)(20,35,38,54,40,75,76,53,39,70,74,52,37,58,44,25)(30,62,64,47,63,68,78,79,36,71,72,51,65,80,67,60);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19)(18,24)(20,36)(21,33)(22,43)(23,34)(25,51)(26,49)(27,57)(28,50)(29,56)(30,39)(31,42)(32,45)(35,68)(37,63)(38,64)(40,65)(41,66)(44,67)(46,73)(47,53)(48,59)(52,79)(54,60)(55,61)(58,62)(69,77)(70,80)(71,75)(72,74)(76,78);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20,23,39)(18,25,28,53)(19,30)(21,40,42,37)(22,44,45,76)(24,47)(26,54,56,52)(27,58,59,75)(29,60)(31,65)(32,67)(33,63)(34,36)(35,69,70,55)(38,73,74,41)(43,78)(46,64)(48,62)(49,79)(50,51)(57,71)(61,80)(66,72)(68,77)

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53262);