#294221

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

745 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,23)(13,20)(14,19)(15,18)(16,17)(22,24)(25,26,29,33)(27,34,37,31)(28,38,30,32)(35,40,36,39);(2,5)(6,7)(9,11,14,17)(10,16,19,21)(12,18,24,20)(13,22,15,23)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(9,12,13,21,14,24,15,16)(10,17,18,22,19,11,20,23)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,18,19,20)(11,22,17,23)(12,16,24,21)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201417);