#141084

order	512
nilpotency class	4
isoclinism	isoclinic class of #141077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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)(17,18,22,29,23,28,38,47,24,30,37,26,21,39,20,27)(19,31,35,45,34,46,33,42,36,41,32,25,40,43,48,44);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19,23,40,24,36,21,34)(18,25,28,42,30,45,39,44)(20,32,22,33,38,35,37,48)(26,41,27,46,29,31,47,43);(2,7)(3,10)(8,12)(11,16)(17,20,21,37,24,38,23,22)(18,26,28,27,30,29,39,47)(19,32,34,48,36,35,40,33)(25,41,42,46,45,31,44,43)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=eg,d^2=ghi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48280);