#137008

order	512
nilpotency class	4
isoclinism	isoclinic class of #136889
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	C₂³	#448
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,9,11,16)(4,6)(5,8)(7,10,14,13)(12,15)(17,18,22,28,37,25,39,26,23,27,36,44,20,29,21,38)(19,30,34,24,31,42,32,46,33,47,48,43,35,45,40,41);(2,6)(5,12)(7,14)(10,13)(17,19,20,31,23,33,37,35)(18,24,25,41,27,43,29,46)(21,34,36,32,39,48,22,40)(26,45,44,47,38,42,28,30);(1,3,4,11)(2,7,6,14)(5,13,12,10)(8,16,15,9)(17,20,23,37)(18,25,27,29)(19,31,33,35)(21,22,39,36)(24,41,43,46)(26,28,38,44)(30,42,47,45)(32,34,40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=h,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh

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, 44204);