#349732

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

837 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)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26)(27,35)(28,31)(29,32)(30,34)(33,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(10,14)(15,23)(16,21)(17,24)(26,31)(27,36)(32,34)(37,38);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(26,32)(29,30)(31,34)(33,40)(35,39)(37,38);(25,27,28,36)(26,33,31,39)(29,38,30,37)(32,40,34,35)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=hi,d^2=i,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, 256928);