#487265

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	#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

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

normal subgroups only

This group has 693 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)(3,11,20,43)(4,22,17,42)(5,12,27,49)(6,28,31,13)(7,33,35,50)(9,16,36,15)(10,29,39,18)(19,47,25,58)(21,45,23,44)(24,60,30,37)(26,63,64,48)(32,52,61,53)(34,57,54,51)(38,62,46,56)(40,55,41,59);(1,3,8,20)(2,10,14,39)(4,7,17,35)(5,25,27,19)(6,29,31,18)(9,21,36,23)(11,13,43,28)(12,46,49,38)(15,40,16,41)(22,57,42,51)(24,54,30,34)(26,56,64,62)(32,59,61,55)(33,37,50,60)(44,52,45,53)(47,63,58,48);(1,4,9,19)(2,11,16,45)(3,7,21,5)(6,30,32,62)(8,17,36,25)(10,13,41,53)(12,47,33,22)(14,43,15,44)(18,54,55,64)(20,35,23,27)(24,61,56,31)(26,29,34,59)(28,40,52,39)(37,57,46,63)(38,48,60,51)(42,49,58,50);(1,5,9,7)(2,12,16,33)(3,17,21,25)(4,23,19,20)(6,26,32,34)(8,27,36,35)(10,37,41,46)(11,42,45,58)(13,48,53,51)(14,49,15,50)(18,30,55,62)(22,44,47,43)(24,59,56,29)(28,63,52,57)(31,64,61,54)(38,39,60,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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,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, 394461);