#142620

order	512
nilpotency class	4
isoclinism	isoclinic class of #142620
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
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 571 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:

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

Polycyclic presentation:

2^9|d^2=gi,e^2=gi,f^2=i,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,f^b=fi,g^b=gi,d^c=dh,e^c=eg,f^c=fi,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, 49816);