#491663

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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

510 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)(17,18,23,27)(19,28,30,36)(20,32,34,46)(21,37,38,25)(22,29,39,45)(24,40,41,31)(26,42,43,33)(35,48,47,44);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,21,31,23,30,38,40)(18,24,25,28,27,41,37,36)(20,29,35,43,34,45,47,26)(22,32,33,44,39,46,42,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,33,24,21,35,39,28,23,34,42,41,38,47,22,36)(18,19,29,44,25,31,43,46,27,30,45,48,37,40,26,32);(3,9)(7,14)(10,15)(12,16)(19,30)(20,34)(24,41)(28,36)(31,40)(32,46)(35,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,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, 398859);