#256723

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

730 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)(11,16)(12,21)(13,18)(14,23)(15,19)(17,20)(22,24)(25,26,29,33)(27,35,38,31)(28,34,30,32)(36,40,37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,13)(14,22)(15,20)(17,19)(18,21)(23,24)(25,27,29,38)(26,31,33,35)(28,37,30,36)(32,39,34,40);(1,4,5,3)(2,6,7,8)(9,12,15,22)(10,17,19,16)(11,13,20,14)(18,24,23,21)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(2,7)(6,8)(9,13,15,14)(10,18,19,23)(11,12,20,22)(16,21,17,24)(25,28,29,30)(26,32,33,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=eh,d^2=h,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=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163919);