#123206

order	512
nilpotency class	4
isoclinism	isoclinic class of #123191
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

270 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 42 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)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,24,30)(19,31,35,59)(20,36,39,56)(21,40,44,54)(22,29,23,28)(25,47,50,33)(26,45,52,38)(27,42,55,43)(32,60,64,77)(34,65,46,53)(37,68,71,51)(41,57,72,49)(48,69,76,80)(58,63,79,62)(61,66,74,75)(67,73,78,70);(3,10)(6,14)(7,12)(11,16)(17,19,24,35)(18,25,30,50)(20,32,39,64)(21,41,44,72)(22,34,23,46)(26,48,52,76)(27,53,55,65)(28,49,29,57)(31,54,59,40)(33,42,47,43)(36,66,56,75)(37,69,71,80)(38,62,45,63)(51,77,68,60)(58,78,79,67)(61,73,74,70);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,20)(18,26)(19,32)(21,37)(22,45)(23,38)(24,39)(25,48)(27,51)(28,56)(29,36)(30,52)(31,58)(33,61)(34,63)(35,64)(40,67)(41,69)(42,73)(43,70)(44,71)(46,62)(47,74)(49,75)(50,76)(53,77)(54,78)(55,68)(57,66)(59,79)(60,65)(72,80)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30402);