#145744

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	#33765	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 316 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)(3,14)(4,8)(5,25)(6,28)(7,13)(9,17)(10,21)(11,40)(12,22)(15,38)(16,52)(18,47)(19,46)(20,35)(23,39)(24,32)(26,43)(27,42)(29,54)(30,62)(31,58)(33,57)(34,64)(36,50)(37,51)(41,48)(44,49)(45,55)(53,61)(56,63)(59,60);(2,8)(3,15)(4,21)(5,26)(6,22)(9,33)(10,12)(11,41)(13,32)(14,44)(16,53)(17,20)(18,51)(19,50)(23,48)(24,58)(25,60)(27,40)(28,31)(29,35)(30,42)(34,47)(36,63)(37,45)(38,52)(39,43)(46,61)(49,55)(54,56)(57,64);(1,3,4,16,22,19,58,56,7,20,24,57,6,18,21,55)(2,9,10,34,28,37,32,44,13,38,31,61,12,36,8,29)(5,17,23,54,40,50,62,53,27,15,48,49,26,51,59,64)(11,14,39,45,25,47,60,33,43,35,41,63,42,46,30,52)

Polycyclic presentation:

2^9|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, 52940);