#411192

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

888 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318388);