#9716841

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	#49190	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34)(35,45)(36,39)(37,42)(38,43)(40,44)(41,47)(46,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,28)(18,24,25,32)(20,29,31,22)(21,27,26,30)(33,35,40,47)(34,41,44,45)(36,46,37,38)(39,43,42,48);(2,7)(6,13)(8,11)(14,15)(18,25)(19,28)(20,31)(27,30)(33,36,40,37)(34,39,44,42)(35,46,47,38)(41,43,45,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,26)(19,29)(21,25)(22,28)(23,31)(24,30)(27,32)(33,37,40,36)(34,42,44,39)(35,38,47,46)(41,48,45,43);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,26)(18,20,25,31)(19,30,28,27)(22,32,29,24)(33,38)(34,43)(35,36)(37,47)(39,45)(40,46)(41,42)(44,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22,23,29)(18,27,25,30)(19,20,28,31)(21,24,26,32)(33,39,40,42)(34,36,44,37)(35,43,47,48)(38,41,46,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9624037);