#134838

order	512
nilpotency class	4
isoclinism	isoclinic class of #134745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#427
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

223 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fi,e^c=ei,f^c=fhi

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, 42034);