#9009229

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=i,f^2=gh,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, 8916425);