#142697

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	#3058	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 337 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,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, 49893);