#10411763

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

4709 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)(9,10)(11,16)(12,18)(13,15)(14,17)(19,23)(20,24)(21,22)(26,30)(27,33)(28,35)(39,40);(1,3,7,6)(2,4,8,5)(10,13)(16,18)(17,21)(23,24)(25,26,29,30)(27,31,33,37)(28,32,35,38)(34,39,36,40);(1,4,7,5)(2,6,8,3)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,39)(35,36)(38,40);(1,4,7,5)(2,3,8,6)(9,12,15,11)(10,16,13,18)(14,19,22,20)(17,24,21,23)(25,27)(26,31)(28,36)(29,33)(30,37)(32,40)(34,35)(38,39);(1,5,7,4)(2,3,8,6)(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40);(1,6,7,3)(2,4,8,5)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,28,29,35)(26,32,30,38)(27,34,33,36)(31,39,37,40)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10318959);