#146289

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

Polycyclic presentation:

2^9|a^2=dhi,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, 53485);