#172262

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

643 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79458);