#138380

order	512
nilpotency class	4
isoclinism	isoclinic class of #138195
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45576);