#137935

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
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	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 356 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(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);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(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)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(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=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dgi,e^b=egi,f^b=fgi,d^c=dgi,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, 45131);