#131752

order	512
nilpotency class	4
isoclinism	isoclinic class of #131648
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,h^b=hi,e^c=eg,f^c=fg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38948);