#138726

order	512
nilpotency class	4
isoclinism	isoclinic class of #138611
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

389 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

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

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

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,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45922);