#145591

order	512
nilpotency class	4
isoclinism	isoclinic class of #145590
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18)(19,24)(20,25)(21,38)(22,39)(23,28)(26,35)(27,36)(29,47)(30,43)(31,48)(32,41)(33,45)(34,42)(37,46)(40,44);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,19,23,33)(18,24,28,45)(20,34,37,31)(21,29,35,30)(22,40,36,32)(25,42,46,48)(26,43,38,47)(27,41,39,44);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(18,25)(19,29)(20,35)(21,37)(22,36)(24,41)(26,27)(28,46)(30,33)(31,40)(32,34)(38,39)(43,47)(44,45)

Polycyclic presentation:

2^9|b^2=i,c^2=d,d^2=g,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fhi,h^b=hi,e^c=egh,f^c=fghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52787);