#487273

order	512
nilpotency class	4
isoclinism	isoclinic class of #486883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi

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, 394469);