#278599

order	512
nilpotency class	3
isoclinism	isoclinic class of #278563
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	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,10,14,16)(4,9,12,7)(6,15,11,13)(17,18,22,26,23,27,21,37)(19,28,31,40,32,45,30,24)(20,33,35,43,36,42,38,25)(29,39,46,48,34,41,47,44);(1,2,4,7,5,8,12,9)(3,6,13,10,14,11,15,16)(17,19,21,30,23,32,22,31)(18,24,26,28,27,40,37,45)(20,29,35,46,36,34,38,47)(25,39,42,44,43,41,33,48);(1,2)(3,11)(4,7)(5,8)(6,14)(9,12)(10,15)(13,16)(17,20,23,36)(18,25,27,43)(19,29,32,34)(21,35,22,38)(24,39,40,41)(26,42,37,33)(28,44,45,48)(30,46,31,47);(1,3,12,15,5,14,4,13)(2,6,9,16,8,11,7,10)(17,19,21,30,23,32,22,31)(18,24,26,28,27,40,37,45)(20,34,35,47,36,29,38,46)(25,41,42,48,43,39,33,44)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185795);