#129387

order	512
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 122 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)(10,15,16,13)(12,14)(17,18,21,26,22,27,36,46,23,28,37,47,38,48,59,69)(19,24,31,41,32,42,55,65,30,40,53,63,54,64,74,78)(20,25,34,44,35,45,58,68,33,43,56,66,57,67,75,79)(29,49,51,70,52,71,73,80,50,39,60,61,72,62,76,77);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,39)(26,41)(27,42)(28,40)(33,50)(34,51)(35,52)(36,55)(37,53)(38,54)(43,49)(44,61)(45,62)(46,65)(47,63)(48,64)(56,60)(57,72)(58,73)(59,74)(66,70)(67,71)(68,77)(69,78)(75,76)(79,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,35,23,33,38,57)(18,25,27,45,28,43,48,67)(19,29,32,52,30,50,54,72)(21,34,36,58,37,56,59,75)(24,39,42,62,40,49,64,71)(26,44,46,68,47,66,69,79)(31,51,55,73,53,60,74,76)(41,61,65,77,63,70,78,80)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^b=ei,f^b=fg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36583);