#10519369

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

4436 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10426565);