#278537

order	512
nilpotency class	3
isoclinism	isoclinic class of #278523
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

511 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)(3,10,13,16)(4,9,11,7)(6,14,15,12)(17,18,23,28,24,29,22,38)(19,27,21,37,32,44,31,25)(20,33,35,43,36,42,39,26)(30,40,45,47,46,48,34,41);(1,2,4,7,5,8,11,9)(3,6,12,10,13,15,14,16)(17,19,22,31,24,32,23,21)(18,25,28,27,29,37,38,44)(20,30,35,45,36,46,39,34)(26,40,42,41,43,48,33,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,20,24,36)(18,26,29,43)(19,30,32,46)(21,34,31,45)(22,35,23,39)(25,40,37,48)(27,41,44,47)(28,42,38,33);(1,3,11,14,5,13,4,12)(2,6,9,16,8,15,7,10)(17,21,23,32,24,31,22,19)(18,27,38,25,29,44,28,37)(20,34,39,46,36,45,35,30)(26,41,33,40,43,47,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=eh,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,f^c=fi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185733);