#9039147

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946343);