#146288

order	512
nilpotency class	4
isoclinism	isoclinic class of #146288
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2101
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#479
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

142 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53484);