#139679

order	512
nilpotency class	4
isoclinism	isoclinic class of #139568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#416
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46875);