#144116

order	512
nilpotency class	4
isoclinism	isoclinic class of #144085
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
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 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi,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, 51312);