#311318

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

573 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)(3,8)(4,6)(5,7)(9,10,15,21)(11,17,23,18)(12,16,22,24)(13,20,14,19)(25,26)(27,34)(28,38)(29,32)(30,33)(31,37)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11,14,22,15,23,13,12)(10,16,20,18,21,24,19,17)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,40)(38,39);(1,4,5,3)(2,6,7,8)(9,12,13,23,15,22,14,11)(10,17,19,24,21,18,20,16)(26,32)(27,35)(28,30)(31,39)(34,40)(36,37);(2,7)(6,8)(9,12,13,23,15,22,14,11)(10,18,19,16,21,17,20,24)(25,28,29,30)(26,33,32,38)(27,36,37,35)(31,40,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=ei,d^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=di,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, 218514);