#487215

order	512
nilpotency class	4
isoclinism	isoclinic class of #486827
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×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 725 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,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, 394411);