#142698

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:

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

Polycyclic presentation:

2^9|a^2=i,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, 49894);