#137912

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,23,28,24,27,41,46,21,29,36,42,22,37,39,47)(19,20,31,35,33,34,26,44,32,25,40,43,30,38,45,48);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,23,40,24,33,41,45,21,32,36,31,22,30,39,26)(18,25,42,43,27,38,47,48,29,20,28,35,37,34,46,44);(1,4,6,14)(2,7,8,12)(3,10,11,16)(5,15,9,13)(17,20,21,25)(18,26,29,45)(19,28,32,42)(22,38,24,34)(23,35,36,43)(27,40,37,31)(30,47,33,46)(39,48,41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=di,e^b=egi,f^b=fi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45108);