#243821

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151017);