#138760

order	512
nilpotency class	4
isoclinism	isoclinic class of #138694
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)(17,18,20,26,22,28,37,43,21,27,36,25,38,45,35,24)(19,29,31,47,33,44,34,23,30,40,32,42,39,46,48,41);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19)(18,23,27,41)(20,34)(21,30)(22,39)(24,29,43,40)(25,44,26,46)(28,47,45,42)(31,37)(32,35)(33,38)(36,48);(3,11)(7,15)(10,16)(12,13)(18,24)(19,30)(20,35)(22,38)(23,40)(25,28)(26,45)(27,43)(29,41)(31,34)(32,48)(36,37)(42,44)(46,47)

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,c^b=cg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,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, 45956);