#411479

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

898 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)(3,8)(4,6)(5,7)(17,18)(19,21)(20,22)(23,24)(25,26,30,32,29,33,28,34)(27,31,37,38,35,40,36,39);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,38)(33,40)(34,39);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(26,32)(28,30)(31,38)(33,34)(36,37)(39,40);(9,11,12,15)(10,14,13,16)(18,22)(21,24)(26,33)(27,35)(32,34)(36,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=ehi,c^2=e,d^2=g,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, 318675);