#135530

order	512
nilpotency class	4
isoclinism	isoclinic class of #135529
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8747
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=dh,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,f^b=fgi,h^b=hi,d^c=di,e^c=eg,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42726);