#137981

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

568 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,5,7,8)(4,6)(9,10)(11,22,27,48)(12,28,31,47)(13,32)(14,20)(15,39)(16,21)(17,25,43,33)(18,30,46,38)(19,35)(23,50)(24,56)(26,52,37,53)(29,45,34,49)(36,40)(41,67)(42,54,44,64)(51,59)(55,63,65,66)(57,69)(58,68)(60,71)(61,62)(70,72);(1,3,4,7)(2,5)(6,8)(9,11,23,54,14,37,36,25,16,27,35,64,15,26,13,33)(10,17,40,53,39,42,50,22,21,43,32,52,20,44,19,48)(12,24,55,72,38,58,29,61,31,60,65,67,30,59,34,57)(18,41,66,56,28,62,45,51,46,70,63,71,47,69,49,68);(9,12)(10,18)(11,24)(13,34)(14,38)(15,30)(16,31)(17,41)(19,49)(20,47)(21,46)(22,51)(23,55)(25,61)(26,59)(27,60)(28,39)(29,36)(32,63)(33,57)(35,65)(37,58)(40,66)(42,62)(43,70)(44,69)(45,50)(48,68)(52,71)(53,56)(54,72)(64,67)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

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, 45177);