#490322

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

652 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)(3,9)(4,11)(8,12)(10,13)(14,16)(17,18)(19,29)(20,34)(21,25)(22,40)(23,28)(24,31)(26,37)(27,38)(30,46)(32,42)(33,47)(35,44)(36,39)(41,45)(43,48);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,19,23,33)(18,24,28,44)(20,35,37,31)(21,30,39,48)(22,41,38,32)(25,29,36,47)(26,45,34,42)(27,46,40,43);(1,3,5,10)(2,7,8,15)(4,13,11,9)(6,14,12,16)(17,18)(19,24)(20,36)(21,26)(22,27)(23,28)(25,37)(29,31)(30,45)(32,43)(33,44)(34,39)(35,47)(38,40)(41,46)(42,48);(2,8)(6,12)(7,15)(14,16)(17,20,22,21,23,37,38,39)(18,25,27,34,28,36,40,26)(19,30,32,35,33,48,41,31)(24,42,43,29,44,45,46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397518);