#143372

order	512
nilpotency class	4
isoclinism	isoclinic class of #143211
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	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50568);