#146116

order	512
nilpotency class	4
isoclinism	isoclinic class of #146116
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

191 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53312);