#145362

order	512
nilpotency class	4
isoclinism	isoclinic class of #145359
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,17,24)(11,25,31,60)(12,32,39,68)(13,21,42,57)(14,20,41,58)(15,23,16,22)(18,43,48,27)(19,49,55,35)(26,61,66,44)(28,46,40,47)(29,45,30,56)(33,69,62,71)(34,53,36,54)(37,51,38,52)(50,63,59,70)(64,67,65,72);(9,11,17,31)(10,18,24,48)(12,33,39,62)(13,40,42,28)(14,27,41,43)(15,30,16,29)(19,50,55,59)(20,56,58,45)(21,25,57,60)(22,47,23,46)(26,34,66,36)(32,67,68,72)(35,64,49,65)(37,70,38,63)(44,51,61,52)(53,69,54,71);(2,5)(3,8)(6,7)(9,12,13,34,16,38,41,49,17,39,42,36,15,37,14,35)(10,19,20,51,23,54,57,68,24,55,58,52,22,53,21,32)(11,26,27,62,30,65,28,63,31,66,43,33,29,64,40,70)(18,44,25,59,47,72,45,71,48,61,60,50,46,67,56,69)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52558);