#145731

order	512
nilpotency class	4
isoclinism	isoclinic class of #145730
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10564
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

306 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gi,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52927);