#146193

order	512
nilpotency class	4
isoclinism	isoclinic class of #143142
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂³	#474
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 226 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=deg,c^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,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, 53389);