#145259

order	512
nilpotency class	4
isoclinism	isoclinic class of #145214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#482
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=dh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,d^c=di,e^c=eghi,f^c=fgi,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52455);