#9545548

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	#49842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,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, 9452744);