#474943

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

964 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382139);