#137247

order	512
nilpotency class	4
isoclinism	isoclinic class of #137153
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,20,28,22,29,35,24,23,30,38,26,21,27,37,41)(19,31,32,47,33,45,48,43,34,46,36,40,39,25,42,44);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(18,24,30,41)(20,35)(21,22)(25,40,45,47)(26,29,28,27)(31,43,46,44)(32,48)(33,39)(36,42)(37,38);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(17,19,23,34)(18,25,30,45)(20,36,38,32)(21,39,22,33)(24,40,41,47)(26,44,28,43)(27,46,29,31)(35,42,37,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44443);