#9206570

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	#53461	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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)(3,8)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,30)(31,32)(33,34,38,43)(35,42,45,39)(36,46,44,40)(37,41,47,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19,23,22)(18,24,27,28)(20,29,26,31)(21,30,25,32)(33,35,38,45)(34,39,43,42)(36,37,44,47)(40,41,46,48);(2,4)(3,9)(10,12)(11,16)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,32,28,30)(33,36,38,44)(34,40,43,46)(35,37,45,47)(39,41,42,48);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,32)(28,30)(33,37,38,47)(34,41,43,48)(35,44,45,36)(39,46,42,40);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,31,28,29)(33,35)(34,42)(36,37)(38,45)(39,43)(40,48)(41,46)(44,47);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,30,25,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=h,e^2=ghi,f^2=gh,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, 9113766);