#9166787

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
prime radical	#47878	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,29)(30,31)(33,34,39,43)(35,42,38,37)(36,41,47,40)(44,45,46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,21)(18,23,25,26)(20,27,28,30)(24,31,29,32)(33,35,39,38)(34,37,43,42)(36,44,47,46)(40,45,41,48);(3,10)(6,7)(11,16)(14,15)(20,28)(24,29)(27,30)(31,32)(33,36,39,47)(34,40,43,41)(35,44,38,46)(37,45,42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,29)(26,32)(33,36)(34,41)(35,44)(37,48)(38,46)(39,47)(40,43)(42,45);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,18)(19,23)(20,29)(21,26)(22,25)(24,28)(27,32)(30,31)(33,37)(34,35)(36,45)(38,43)(39,42)(40,44)(41,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,19)(18,23,25,26)(20,30,28,27)(24,31,29,32)(33,38,39,35)(34,42,43,37)(36,46,47,44)(40,48,41,45)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9073983);