#138565

order	512
nilpotency class	4
isoclinism	isoclinic class of #138564
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 296 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 45761);