#170513

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37202	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

743 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,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, 77709);