#142702

order	512
nilpotency class	4
isoclinism	isoclinic class of #142700
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

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

normal subgroups only

This group has 338 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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=dgi,e^b=eg,f^b=fi,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, 49898);