#144069

order	512
nilpotency class	4
isoclinism	isoclinic class of #143959
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	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 373 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 51265);