#490065

order	512
nilpotency class	4
isoclinism	isoclinic class of #489733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61443	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

686 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fhi,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397261);