#9729210

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9636406);