#350209

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 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)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(16,24)(17,21)(22,23)(25,26,28,34)(27,35,36,31)(29,32,30,33)(37,39,38,40);(2,5)(6,8)(10,15)(14,20)(16,23)(22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,20)(13,15)(14,19)(17,22)(18,23)(21,24)(26,32)(27,36)(29,30)(31,39)(33,34)(35,40);(9,11,13,18)(10,16,15,23)(12,21,19,17)(14,22,20,24)(26,33)(29,30)(31,39)(32,34)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,c^2=ei,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ch,f^c=fh,g^c=gh,f^d=fh,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, 257405);