#145021

order	512
nilpotency class	4
isoclinism	isoclinic class of #145020
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10409
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

371 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dg,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egi,f^c=fh,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, 52217);