#9009201

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	#46790	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(33,34,39,43)(35,37,44,40)(36,41,46,48)(38,42,47,45);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,35)(34,40)(36,38)(37,43)(39,44)(41,45)(42,48)(46,47);(2,8)(4,7)(5,14)(11,16)(18,22)(19,25)(20,27)(31,32)(33,36,39,46)(34,41,43,48)(35,38,44,47)(37,42,40,45);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19,21,25)(18,23,22,29)(20,28,27,26)(24,32,30,31)(33,37,39,40)(34,35,43,44)(36,45,46,42)(38,41,47,48);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,20,21,27)(18,24,22,30)(19,26,25,28)(23,31,29,32)(33,36)(34,41)(35,38)(37,42)(39,46)(40,45)(43,48)(44,47);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,38)(34,42)(35,36)(37,41)(39,47)(40,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8916397);