#143887

order	512
nilpotency class	4
isoclinism	isoclinic class of #143815
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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 ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

361 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 66 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)(19,27)(20,31)(21,36)(22,25)(23,29)(24,34)(26,33)(28,43)(30,45)(32,42)(35,46)(37,40)(38,44)(39,47)(41,48);(1,3,6,4)(2,7,10,16)(5,13,15,12)(8,14,11,9)(23,38)(27,43)(39,42)(40,45);(1,4,6,3)(2,8,10,11)(5,12,15,13)(7,9,16,14)(17,19)(18,23)(20,32)(21,37)(22,28)(24,40)(25,38)(26,42)(27,31)(29,34)(30,33)(35,47)(36,39)(41,45)(43,46)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51083);