#131846

order	512
nilpotency class	4
isoclinism	isoclinic class of #131844
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8443
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#61105	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

787 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=h,e^2=g,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39042);