#487217

order	512
nilpotency class	4
isoclinism	isoclinic class of #486827
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×C₂²⋊C₄	C₂⁴
prime radical	#392	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 709 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,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, 394413);