#138732

order	512
nilpotency class	4
isoclinism	isoclinic class of #138616
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)(9,10,14,21,39,42,37,19,15,22,38,53,16,23,13,20)(11,24,28,17,40,44,26,54,29,57,64,45,30,51,27,46)(12,31,33,49,41,52,32,50,34,47,36,18,35,48,66,55)(25,43,60,68,65,71,59,67,61,69,63,56,62,70,58,72);(2,6,8,7)(9,11,16,40,15,29,39,30)(10,17,42,46,22,45,23,54)(12,25,41,62,34,61,35,65)(13,28,38,26,37,64,14,27)(18,43,50,71,49,69,55,70)(19,51,53,57,20,44,21,24)(31,56,48,72,47,68,52,67)(32,63,36,59,66,60,33,58);(9,12)(10,18)(11,25)(13,36)(14,32)(15,34)(16,41)(17,43)(19,48)(20,52)(21,31)(22,49)(23,55)(24,56)(26,60)(27,63)(28,59)(29,61)(30,65)(33,37)(35,39)(38,66)(40,62)(42,50)(44,67)(45,69)(46,71)(47,53)(51,72)(54,70)(57,68)(58,64)

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,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,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, 45928);