#141395

order	512
nilpotency class	4
isoclinism	isoclinic class of #141346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 336 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ehi,d^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48591);