#305140

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

731 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=efi,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=dh,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, 212336);