#139655

order	512
nilpotency class	4
isoclinism	isoclinic class of #139598
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#534	C₂×C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

449 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46851);