#145866

order	512
nilpotency class	4
isoclinism	isoclinic class of #145866
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10575
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#455
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

259 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,8)(4,9,19,35)(5,22)(6,10)(7,28)(11,39)(12,25)(13,45,46,58)(14,31)(15,40)(16,43)(17,33)(18,52,53,34)(20,41,55,64)(21,44,29,37)(23,32)(24,36,56,63)(26,42)(27,38)(30,47,57,51)(48,60,54,62)(49,61)(50,59);(1,3)(2,8)(4,18,19,53)(5,23)(6,17)(7,14)(9,34,35,52)(10,33)(11,40)(12,43)(13,20,46,55)(15,39)(16,25)(21,48,29,54)(22,32)(24,51,56,47)(26,49)(27,50)(28,31)(30,36,57,63)(37,60,44,62)(38,59)(41,45,64,58)(42,61);(1,4,7,29,6,19,25,21)(2,9,12,44,10,35,28,37)(3,13,16,51,17,46,14,47)(5,24,27,55,26,56,39,20)(8,30,31,45,33,57,43,58)(11,41,42,63,38,64,22,36)(15,48,49,53,50,54,23,18)(32,60,59,52,61,62,40,34)

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dh,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=eg,f^c=fhi,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, 53062);