#145564

order	512
nilpotency class	4
isoclinism	isoclinic class of #145563
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10539
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
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

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fhi,g^b=gi,e^c=eghi,f^c=fghi,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, 52760);