#144012

order	512
nilpotency class	4
isoclinism	isoclinic class of #143942
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	#3147	C₂²
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

353 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 78 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)(17,18,22,28)(19,30,33,45)(20,29,23,26)(21,36,39,46)(24,34,43,31)(25,37,42,32)(27,35,41,40)(38,47,48,44);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,22,33)(18,24,28,43)(20,34,23,31)(21,37,39,32)(25,41,42,27)(26,45,29,30)(35,38,40,48)(36,44,46,47);(3,12)(4,5)(7,11)(8,9)(13,14)(15,16)(17,19)(18,25)(20,31)(21,38)(22,33)(23,34)(24,41)(26,44)(27,43)(28,42)(29,47)(30,36)(32,40)(35,37)(39,48)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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=fg,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,d^c=dgi,e^c=eh,f^c=fgi,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, 51208);