#10047452

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

4439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9954648);