#141530

order	512
nilpotency class	4
isoclinism	isoclinic class of #141454
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 292 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=ghi,e^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48726);