#9545547

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	#48168	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452743);