#170963

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#7742	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

825 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 78159);