#144104

order	512
nilpotency class	4
isoclinism	isoclinic class of #144085
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 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,23,28)(19,30,33,42)(20,24,21,37)(22,40,36,29)(25,43,44,31)(26,38,27,34)(32,46,41,47)(35,45,39,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,24)(20,34)(21,38)(22,36)(25,30)(26,29)(27,40)(28,37)(31,39)(32,41)(35,43)(42,44)(45,46)(47,48);(4,14)(6,15)(7,12)(10,13)(17,19)(18,25)(20,35)(21,39)(22,41)(23,33)(24,42)(26,46)(27,47)(28,44)(29,48)(30,37)(31,38)(32,36)(34,43)(40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi,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, 51300);