#142696

order	512
nilpotency class	4
isoclinism	isoclinic class of #142696
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9950
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#33765	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

345 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,16)(4,22)(5,10)(6,13)(7,14)(8,15)(9,18)(11,26)(12,29)(17,39)(19,51)(20,46)(21,47)(23,34)(24,42)(25,57)(27,41)(28,45)(30,37)(31,38)(32,61)(33,60)(35,49)(36,58)(40,59)(43,50)(44,48)(52,62)(53,64)(54,56)(55,63);(1,3,7,17,8,21,6,20)(2,9,13,34,15,35,14,44)(4,23,24,18,27,48,26,49)(5,28,30,52,31,53,29,19)(10,33,12,43,38,63,37,32)(11,39,41,16,42,46,22,47)(25,50,56,60,59,61,58,55)(36,62,40,45,54,51,57,64);(2,10)(3,17)(4,24)(5,29)(6,7)(9,32)(11,40)(12,14)(13,37)(15,38)(16,45)(18,49)(19,52)(20,21)(22,54)(26,27)(30,31)(33,44)(34,63)(35,43)(36,41)(39,51)(42,57)(46,62)(47,64)(50,55)(56,58)(60,61)

Polycyclic presentation:

2^9|b^2=h,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=dgi,e^b=eg,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, 49892);