#144451

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

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

499 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=i,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, 51647);