#145355

order	512
nilpotency class	4
isoclinism	isoclinic class of #145355
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10472
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33594	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,15)(4,21)(5,28)(6,12)(7,11)(8,32)(9,17)(10,24)(13,30)(14,29)(16,47)(18,36)(19,46)(20,45)(22,54)(23,39)(25,40)(26,42)(27,41)(31,44)(33,43)(34,48)(35,52)(37,51)(38,49)(50,64)(53,58)(55,62)(56,61)(57,59)(60,63);(1,3)(2,9)(4,22)(5,19)(6,20)(7,17)(8,33)(10,38)(11,43)(12,37)(13,45)(14,46)(15,28)(16,25)(18,29)(21,47)(23,49)(24,55)(26,48)(27,56)(30,52)(31,51)(32,35)(34,40)(36,44)(39,54)(41,64)(42,63)(50,57)(53,61)(58,60)(59,62);(1,4,5,23,29,24,31,57,6,27,30,58,8,26,7,25)(2,10,11,39,32,21,13,40,12,42,44,53,14,41,28,59)(3,16,17,48,33,60,52,56,20,50,51,55,18,49,19,22)(9,34,15,47,46,54,36,38,37,62,45,64,35,61,43,63)

Polycyclic presentation:

2^9|c^2=d,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,e^c=egh,h^c=hi

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, 52551);