#138750

order	512
nilpotency class	4
isoclinism	isoclinic class of #138689
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45946);