#487260

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,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, 394456);