#303744

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 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)(9,10,14,17)(11,15,19,22)(12,21,16,13)(18,24,23,20)(25,26,29,31)(27,35,37,40)(28,34,30,33)(32,36,39,38);(1,3,5,4)(2,6,8,7)(9,11,14,19)(10,15,17,22)(12,18,16,23)(13,20,21,24)(26,31)(27,36)(28,30)(32,35)(37,38)(39,40);(1,3,5,4)(2,7,8,6)(9,12,13,10,14,16,21,17)(11,18,20,15,19,23,24,22)(25,27,30,36,29,37,28,38)(26,32,34,40,31,39,33,35);(1,4,5,3)(2,7,8,6)(11,19)(15,22)(18,23)(20,24)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,40,39,35)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210940);