#487287

order	512
nilpotency class	4
isoclinism	isoclinic class of #486867
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 686 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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=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, 394483);