#143127

order	512
nilpotency class	4
isoclinism	isoclinic class of #143090
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 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(18,23)(19,28)(20,32)(21,34)(24,40)(25,33)(26,38)(27,39)(29,45)(30,47)(31,44)(35,36)(37,41)(42,46);(6,15)(9,16)(17,18,22,27)(19,29,31,24)(20,33,35,38)(21,25,34,26)(23,36,39,32)(28,42,44,41)(30,40,47,45)(37,43,46,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,21,28,22,31,34,44)(18,24,26,42,27,29,25,41)(20,30,32,43,35,47,36,48)(23,37,33,45,39,46,38,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fh,g^a=gi,c^b=cf,d^b=dg,g^b=gi,d^c=dh,e^c=eh,f^c=fi

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, 50323);