#150976

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#2215	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fi,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58172);