#137844

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	#3100	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 331 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45040);