#145303

order	512
nilpotency class	4
isoclinism	isoclinic class of #145282
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#482
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,4)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,18,20,25)(19,28,33,47)(21,35,34,48)(22,27,23,40)(24,41,42,30)(26,38,36,39)(29,45,44,37)(31,46,32,43);(1,3,6,11)(2,7,9,12)(4,8,13,14)(5,15,16,10)(17,18)(19,29)(20,25)(21,36)(22,40)(23,27)(24,31)(26,34)(28,45)(30,43)(32,42)(33,44)(35,38)(37,47)(39,48)(41,46);(1,4,5,12,6,13,16,7)(2,3,8,10,9,11,14,15)(17,19,22,32,20,33,23,31)(18,24,40,29,25,42,27,44)(21,37,38,30,34,45,39,41)(26,43,48,28,36,46,35,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fi,d^c=di,e^c=eghi,f^c=fgi,h^c=hi

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, 52499);