#487297

order	512
nilpotency class	4
isoclinism	isoclinic class of #486875
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

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

normal subgroups only

This group has 677 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:

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

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,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,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, 394493);