#140548

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#408
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 106 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)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,22,26,23,27,20,34,24,28,35,41,21,37,36,46)(19,25,31,38,32,39,29,42,33,40,43,47,30,45,44,48);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19,22,31,23,32,20,29,24,33,35,43,21,30,36,44)(18,25,26,38,27,39,34,42,28,40,41,47,37,45,46,48);(2,7)(3,10)(8,12)(11,16)(17,20,21,22,24,36,23,35)(18,26,27,34,28,41,37,46)(19,29,30,31,33,44,32,43)(25,38,39,42,40,47,45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47744);