#10042568

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

4515 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)(3,4)(5,6)(7,8)(11,18)(16,19)(20,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(1,3,6,7)(2,4,5,8)(9,10,14,15)(11,19,18,16)(12,13,17,22)(20,24,23,21)(26,31)(28,34)(32,39)(35,40);(1,4,6,8)(2,7,5,3)(10,15)(12,17)(16,19)(20,23)(25,27)(26,32)(28,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,2,6,5)(3,8,7,4)(9,11,14,18)(10,16,15,19)(12,20,17,23)(13,21,22,24)(25,28,30,34)(26,33,31,29)(27,35,37,40)(32,36,39,38);(1,4)(2,3)(5,7)(6,8)(9,12)(10,13)(11,20)(14,17)(15,22)(16,21)(18,23)(19,24)(25,29,30,33)(26,34,31,28)(27,36,37,38)(32,35,39,40);(1,5,6,2)(3,8,7,4)(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,24)(19,20)(25,27)(26,32)(28,35)(29,38)(30,37)(31,39)(33,36)(34,40)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gi,e^2=g,f^2=h,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, 9949764);