#487253

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 ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 701 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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=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, 394449);