#278663

order	512
nilpotency class	3
isoclinism	isoclinic class of #278635
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

497 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)(17,18,23,28,24,29,22,39)(19,30,32,42,33,27,21,25)(20,34,36,45,37,44,40,26)(31,41,38,46,47,48,35,43);(1,2,4,7,5,8,12,9)(3,11,13,16,14,6,15,10)(17,19,22,21,24,33,23,32)(18,25,28,30,29,42,39,27)(20,31,36,38,37,47,40,35)(26,41,44,43,45,48,34,46);(1,2)(3,11)(4,7)(5,8)(6,14)(9,12)(10,15)(13,16)(17,20,24,37)(18,26,29,45)(19,31,33,47)(21,38,32,35)(22,36,23,40)(25,41,42,48)(27,46,30,43)(28,44,39,34);(1,3,12,15,5,14,4,13)(2,6,9,16,8,11,7,10)(17,21,22,33,24,32,23,19)(18,27,28,25,29,30,39,42)(20,35,36,31,37,38,40,47)(26,43,44,48,45,46,34,41)

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,d^b=dh,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, 185859);