#9214722

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

4160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9121918);