#135517

order	512
nilpotency class	4
isoclinism	isoclinic class of #135516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8746
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

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

normal subgroups only

This group has 459 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fg,h^b=hi,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, 42713);