#278314

order	512
nilpotency class	3
isoclinism	isoclinic class of #278299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fi,g^c=gh

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, 185510);