#138518

order	512
nilpotency class	4
isoclinism	isoclinic class of #138488
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45714);