#145745

order	512
nilpotency class	4
isoclinism	isoclinic class of #145744
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

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

normal subgroups only

This group has 308 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=d,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,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,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, 52941);