#9498407

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

4227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9405603);