#145245

order	512
nilpotency class	4
isoclinism	isoclinic class of #145214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
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

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

normal subgroups only

This group has 252 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:

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

Polycyclic presentation:

2^9|a^2=gi,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=fi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,d^c=di,e^c=eghi,f^c=fgi,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52441);