#146179

order	512
nilpotency class	4
isoclinism	isoclinic class of #146179
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10656
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
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

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

normal subgroups only

This group has 179 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=d,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,g^c=gi,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, 53375);