#179785

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

843 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dgi,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, 86981);