#139043

order	512
nilpotency class	4
isoclinism	isoclinic class of #139022
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#466	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=ghi,e^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=egh,f^c=fhi,h^c=hi

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, 46239);