#9545549

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	#48071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3849 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 9452745);