#9160754

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	#50323	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9067950);