#146099

order	512
nilpotency class	4
isoclinism	isoclinic class of #146090
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2071
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=gi,c^2=d,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=eg,f^c=fgh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53295);