#137458

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1879
prime radical	#466	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 74 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)(17,18,22,29,24,31,21,26,25,32,20,28,23,30,40,43)(19,33,36,27,38,42,41,46,39,48,34,47,37,44,35,45);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(18,26)(20,21)(22,40)(23,24)(27,42)(28,30)(29,31)(32,43)(33,46)(34,41)(35,36)(37,38)(44,47)(45,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,19)(18,27)(20,35)(21,36)(22,41)(23,38)(24,37)(25,39)(26,42)(28,33)(29,48)(30,46)(31,45)(32,47)(34,40)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,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, 44654);