#145356

order	512
nilpotency class	4
isoclinism	isoclinic class of #144056
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1697
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10472
prime radical	#33594	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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)(3,14)(4,20)(5,25)(6,29)(7,33)(8,13)(9,16)(10,23)(11,28)(12,27)(15,45)(17,49)(18,37)(19,46)(21,35)(22,40)(24,54)(26,44)(30,36)(31,39)(32,43)(34,42)(38,48)(41,57)(47,64)(50,62)(51,55)(52,58)(53,60)(56,63)(59,61);(1,3)(2,9)(4,21)(5,26)(6,30)(7,34)(8,19)(10,38)(11,42)(12,44)(13,37)(14,25)(15,22)(16,28)(17,27)(18,32)(20,51)(23,59)(24,50)(29,46)(31,48)(33,36)(35,40)(39,62)(41,45)(43,49)(47,60)(52,61)(53,63)(54,64)(55,58)(56,57);(1,4,5,22,27,57,32,60,8,24,6,31,7,23,28,58)(2,10,11,39,33,54,29,53,13,41,43,40,12,20,25,52)(3,15,16,21,34,55,30,59,19,48,18,50,17,47,26,56)(9,35,14,45,44,63,49,64,37,62,46,38,36,61,42,51)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52552);