#136774

order	512
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16)(17,18,22,29,24,30,45,59,23,27,44,46,21,28,41,56)(19,25,34,49,33,48,65,75,32,50,42,57,39,54,70,77)(20,35,38,68,37,53,69,58,36,55,43,71,40,26,51,52)(31,60,62,78,64,47,72,74,63,67,66,80,61,73,79,76);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10)(17,19,22,42,24,33,45,70,23,32,44,34,21,39,41,65)(18,25,46,49,30,48,56,75,27,50,29,57,28,54,59,77)(20,31,43,62,37,64,51,72,36,63,38,66,40,61,69,79)(26,47,52,76,35,67,68,78,53,73,58,74,55,60,71,80);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)(17,20)(18,26)(19,31)(21,40)(22,43)(23,36)(24,37)(25,47)(27,53)(28,55)(29,58)(30,35)(32,63)(33,64)(34,66)(38,44)(39,61)(41,69)(42,62)(45,51)(46,52)(48,67)(49,76)(50,73)(54,60)(56,68)(57,74)(59,71)(65,79)(70,72)(75,78)(77,80)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=ei,f^c=fg

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, 43970);