#487274

order	512
nilpotency class	4
isoclinism	isoclinic class of #486883
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	#392	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

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

normal subgroups only

This group has 692 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,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,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, 394470);