#490898

order	512
nilpotency class	4
isoclinism	isoclinic class of #490891
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	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

428 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)(17,18,22,27)(19,23,28,38)(20,31,35,47)(21,36,33,26)(24,41,42,29)(25,44,45,34)(30,46,37,40)(32,39,48,43);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,31)(25,39)(26,40)(27,38)(32,34)(33,37)(35,41)(36,46)(42,47)(43,45)(44,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,32)(21,37)(22,28)(23,31)(25,40)(26,39)(27,42)(29,34)(30,33)(35,48)(36,43)(38,47)(41,44)(45,46);(3,9)(7,14)(10,15)(12,16)(19,28)(23,38)(24,42)(29,41)(30,37)(32,48)(39,43)(40,46)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398094);