#145976

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,13,18,36,19,22,16,12,20,9,34,11,17,10,37)(7,21,40,45,62,54,56,42,27,55,24,14,26,43,25,44)(8,28,30,51,39,49,32,48,31,52,38,47,33,15,35,50)(23,57,58,68,64,46,29,66,59,65,63,41,60,53,61,67);(1,2)(3,4)(5,7,22,56,11,26,13,40,12,27,10,25,36,62,9,24)(6,14,34,54,19,44,37,55,20,45,18,43,17,42,16,21)(8,29,35,64,33,58,38,23,31,61,32,60,39,63,30,59)(15,46,50,66,28,65,51,41,49,53,48,67,52,57,47,68);(2,4)(5,8,12,31)(6,15)(7,23,27,59)(9,35,13,32)(10,38,22,30)(11,39,36,33)(14,41)(16,50)(17,28)(18,51)(19,52)(20,49)(21,53)(24,61,40,29)(25,63,56,58)(26,64,62,60)(34,47)(37,48)(42,67)(43,57)(44,66)(45,68)(46,55)(54,65)

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,c^2=h,d^2=gi,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^c=dg,e^c=eh,f^c=fgi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53172);