#139417

order	512
nilpotency class	4
isoclinism	isoclinic class of #139356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 80 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)(19,30,35,56)(20,36,39,55)(21,40)(22,28)(23,47)(24,29)(25,33,51,41)(26,38,54,46)(27,43)(31,58)(32,64)(34,60,45,61)(37,53,42,57)(44,48)(49,75)(50,62,52,72)(59,67)(63,71,73,74)(65,77)(66,76)(68,79)(69,70)(78,80);(1,3,5,8,6,9,15,2)(4,10,14,11,12,16,13,7)(17,19,31,62,22,45,44,33,24,35,43,72,23,34,21,41)(18,25,48,61,47,50,58,30,29,51,40,60,28,52,27,56)(20,32,63,80,46,66,37,69,39,68,73,75,38,67,42,65)(26,49,74,64,36,70,53,59,54,78,71,79,55,77,57,76);(1,4,6,12)(2,7,8,11)(3,10,9,16)(5,14,15,13)(17,20,24,39)(18,26,29,54)(19,32,35,68)(21,42,44,37)(22,46,23,38)(25,49,51,78)(27,57,58,53)(28,55,47,36)(30,59,56,76)(31,63,43,73)(33,69,41,65)(34,67,45,66)(40,71,48,74)(50,70,52,77)(60,79,61,64)(62,80,72,75)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46613);