#487291

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 700 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 394487);