#350223

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

852 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)(11,15)(16,21)(17,19)(18,23)(22,24)(25,26,29,33)(27,30,35,39)(28,31,36,32)(34,37,40,38);(2,5)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(1,3,4,7)(2,6,5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=e,d^2=h,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, 257419);