#136736

order	512
nilpotency class	4
isoclinism	isoclinic class of #136735
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43932);