#142617

order	512
nilpotency class	4
isoclinism	isoclinic class of #142616
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9946
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#33531	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,d^c=dh,e^c=eg,g^c=gi,h^c=hi,f^d=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, 49813);