#145813

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

343 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dh,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,g^c=gi

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, 53009);