#139923

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

442 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47119);