#144006

order	512
nilpotency class	4
isoclinism	isoclinic class of #143939
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	C₂³	#469
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

350 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=gi,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,e^b=ei,f^b=fgi,d^c=dg,e^c=eh,f^c=fgi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51202);