#319623

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 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)(3,7)(4,6)(5,8)(9,10,12,16,13,17,20,15)(11,14,19,23,18,22,24,21)(25,26,31,34,30,35,29,39)(27,36,37,33,28,32,38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27,29,38,30,28,31,37)(26,32,39,33,35,36,34,40);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,28,30,27)(26,33,35,40)(29,38,31,37)(32,34,36,39);(2,6)(5,7)(11,18)(14,22)(19,24)(21,23)(25,29,30,31)(26,34,35,39)(27,37,28,38)(32,33,36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226819);