#9498454

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

4172 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)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3)(2,8)(4,13)(5,7)(6,14)(9,11)(10,16)(12,15)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,9)(3,5)(4,15)(6,16)(18,23)(20,29)(21,25)(26,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,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, 9405650);