#138363

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

547 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45559);