#9206747

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3886 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=i,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, 9113943);