#144034

order	512
nilpotency class	4
isoclinism	isoclinic class of #143955
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	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51230);