#138754

order	512
nilpotency class	4
isoclinism	isoclinic class of #138689
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 355 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=h,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45950);