#148053

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^c=di,e^c=eh,f^c=fi,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, 55249);