#491040

order	512
nilpotency class	4
isoclinism	isoclinic class of #491025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,d^b=dgi,d^c=dg,f^c=fh,h^c=hi

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, 398236);