#487285

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	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 683 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:

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

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,c^b=ci,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, 394481);