#9481492

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388688);