#129388

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
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	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 570 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)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,31,35,59,33,47,65,55,34,52,42,70,39,25,45,48)(20,26,37,50,36,49,66,75,38,51,68,77,40,53,69,78)(32,60,63,80,64,46,72,74,62,58,67,79,61,73,71,76);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,25)(20,32)(21,39)(22,42)(23,33)(24,34)(26,46)(27,52)(28,55)(29,31)(30,47)(35,44)(36,64)(37,67)(38,62)(40,61)(41,65)(43,45)(48,57)(49,58)(50,76)(51,73)(53,60)(54,59)(56,70)(63,68)(66,71)(69,72)(74,77)(75,80)(78,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,36,24,38,21,40)(18,26,29,49,30,51,27,53)(19,32,33,64,34,62,39,61)(22,37,43,66,44,68,41,69)(25,46,31,58,47,73,52,60)(28,50,56,75,57,77,54,78)(35,63,65,72,42,67,45,71)(48,74,59,79,55,76,70,80)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,e^b=ei,f^b=fgi,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, 36584);