#349719

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256915);