#136702

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

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,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43898);