#148403

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,d^c=dh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55599);