#138429

order	512
nilpotency class	4
isoclinism	isoclinic class of #138406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=i,d^2=ghi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg,f^c=fg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45625);