#349775

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=i,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, 256971);