#10504529

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

4381 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=gi,f^2=g,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, 10411725);