#146126

order	512
nilpotency class	4
isoclinism	isoclinic class of #146116
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#485
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

174 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,21,18)(12,27,15,26)(16,22)(17,23)(19,25)(20,29,31,28)(30,32)(33,34,36,41,49,59,63,58,38,43,50,57,37,42,48,40)(35,39,45,52,46,53,62,64,47,55,61,51,44,54,60,56);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,18,19,13)(8,20,23,31)(9,24,25,11)(16,28,30,29)(22,27,32,26)(33,35)(34,39)(36,45)(37,44)(38,47)(40,56)(41,52)(42,54)(43,55)(46,49)(48,60)(50,61)(51,57)(53,59)(58,64)(62,63);(1,4,5,16,6,17,19,30)(2,8,9,22,10,23,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27)(34,40)(35,44)(36,48)(37,49)(39,51)(41,42)(43,58)(45,61)(46,47)(50,63)(52,55)(53,64)(54,56)(57,59)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53322);