#491671

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33773	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

527 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)(19,21)(20,31)(22,30)(23,29)(24,27)(25,34)(26,37)(28,43)(32,42)(33,46)(35,47)(36,44)(38,39)(40,48)(41,45);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,21,18,24,30,22,27)(20,28,36,25,35,40,33,41)(23,29,32,26,38,39,37,42)(31,45,46,48,47,34,44,43);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,32,43,22,33,23,34,24,35,37,48,21,36,38,45)(18,25,39,46,19,28,26,31,27,41,29,44,30,40,42,47);(3,9)(7,14)(10,15)(12,16)(17,21,24,22)(18,19,27,30)(20,33,35,36)(23,32,38,37)(25,40,41,28)(26,29,42,39)(31,46,47,44)(34,48,45,43)

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,d^2=i,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=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398867);