#142928

order	512
nilpotency class	4
isoclinism	isoclinic class of #142831
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	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,23,28)(19,29,32,47)(20,26,38,45)(21,39,36,25)(22,40,37,27)(24,30,42,31)(33,44,35,46)(34,48,41,43);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,23,32)(18,24,28,42)(20,35,38,33)(21,30,36,31)(22,41,37,34)(25,43,39,48)(26,29,45,47)(27,46,40,44);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(19,30)(20,36)(21,38)(22,37)(24,29)(25,26)(27,40)(31,32)(33,41)(34,35)(39,45)(42,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50124);