#150790

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,32,34,53)(20,38)(21,45)(22,49)(23,51)(24,31)(25,33,54,67)(26,42)(27,41)(28,48)(29,40)(30,47)(35,70,36,71)(37,72,52,66)(39,63,68,64)(43,79)(44,59)(46,58,50,61)(55,80,56,75)(57,74,65,69)(60,78)(62,76)(73,77);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19,24,34)(18,25,31,54)(20,39,44,68)(21,46,22,50)(23,52,47,37)(26,58,27,61)(28,63,29,64)(30,65,51,57)(32,38,53,59)(33,41,67,42)(35,40,36,48)(43,80,77,75)(45,55,49,56)(60,71,62,70)(66,79,72,73)(69,76,74,78);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,40,76,23,43,21,41,24,44,48,78,47,77,22,42)(18,26,49,73,30,60,28,59,31,27,45,79,51,62,29,38)(19,33,50,75,37,69,35,68,34,67,46,80,52,74,36,39)(25,53,64,70,57,72,55,61,54,32,63,71,65,66,56,58)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fhi,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=cfi,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57986);